Learning Objectives
In this section, you will learn to:
- Write sample spaces.
- Calculate probabilities by examining simple events in sample spaces.
Prerequisite Skills
Before you get started, take this prerequisite quiz.
1. In a city election, there are 2 candidates for mayor, and 3 for supervisor. Use a tree diagram to find the number of ways to fill the two offices.
- Click here to check your answer
The offices can be filled in 6 different ways.
If you missed this problem, review Section 5.2. (Note that this will open in a new window.)
2. For lunch, a small restaurant offers 2 types of soups, three kinds of sandwiches, and two types of soft drinks. Use a tree diagram to determine the number of possible meals consisting of a soup, sandwich, and a soft drink.
- Click here to check your answer
There are 12 different possible meals.
If you missed this problem, review Section 5.2. (Note that this will open in a new window.)
If two coins are tossed, what is the probability that both coins will fall heads? The problem seems simple enough, but it is not uncommon to hear the incorrect answer 1/3. A student may incorrectly reason that if two coins are tossed there are three possibilities, one head, two heads, or no heads. Therefore, the probability of two heads is one out of three. The answer is wrong because if we toss two coins there are four possibilities and not three. For clarity, assume that one coin is a penny and the other a nickel. Then we have the following four possibilities.
HH HT TH TT
The possibility HT, for example, indicates a head on the penny and a tail on the nickel, while TH represents a tail on the penny and a head on the nickel. It is for this reason, we emphasize the need for understanding sample spaces.
Sample Spaces
An act of flipping coins, rolling dice, drawing cards, or surveying people are referred to as a probability experiment. A sample space of an experiment is the set of all possible outcomes.
Example (PageIndex{1})
A single die is rolled. Write the sample space.
Solution
A die has six faces each having an equally likely chance of appearing. Therefore, the set of all possible outcomes (S) is
{ 1, 2, 3, 4, 5, 6 }.
Example (PageIndex{2})
A family has three children. Write the sample space showing the birth order with respect to gender.
Solution
The sample space consists of eight possibilities.
{ BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG }
The possibility BGB, for example, indicates that the first born is a boy, the second born a girl, and the third a boy.
We illustrate these possibilities with a tree diagram.
Example (PageIndex{3})
Two dice are rolled. Write the sample space.
Solution
Let's suppose one of the dice is red, and the other green. We have the following 36 possibilities.
Green | ||||||
Red | 1 | 2 | 3 | 4 | 5 | 6 |
1 | (1, 1) | (1, 2) | (1, 3) | (1, 4) | (1, 5) | (1, 6) |
2 | (2, 1) | (2, 2) | (2, 3) | (2, 4) | (2, 5) | (2, 6) |
3 | (3, 1) | (3, 2) | (3, 3) | (3, 4) | (3, 5) | (3, 6) |
4 | (4, 1) | (4, 2) | (4, 3) | (4, 4) | (4, 5) | (4, 6) |
5 | (5, 1) | (5, 2) | (5, 3) | (5, 4) | (5, 5) | (5, 6) |
6 | (6, 1) | (6, 2) | (6, 3) | (6, 4) | (6, 5) | (6, 6) |
The entry (2, 5), for example, indicates that the red die shows a 2, and the green a 5.
Probability
Now that we understand the concept of a sample space, we will define probability.
An event is a subset of a sample space. For example, in the experiment of rolling two dice, an event might be rolling a sum of 5.
Definition: Probability
The probability of an event describes the chance or likelihood of that event occurring.
For a sample space (S), and an event (A),
- (P(A) = dfrac{ ext{number of ways A appears in S}}{ ext{total number of outcomes in S}} )
- (0 ≤ P(A) ≤ 1)
- The sum of the probabilities of all the outcomes in (S) equals 1.
The probability (P(A)) of an event (A) describes the chance or likelihood of that event occurring.
- If (P(A) = 0), event A is certain not to occur. If (P(A) = 1), event (A) is certain to occur.
- If (P(A) = 0.5), then event A is equally likely to occur or not occur.
- If we toss a fair coin that is equally likely to land on heads or tails, then P(Head) = 0.50.
- If the weather forecast says there is a 70% chance of rain today, then P(Rain) = 0.70, indicating is it more likely to rain than to not rain.
Example (PageIndex{4})
If two dice, one red and one green, are rolled, find the probability that the red die shows a 3 and the green shows a six.
Solution
Since two dice are rolled, there are 36 possibilities. The probability of each outcome, listed in Example (PageIndex{3}), is equally likely.
Since (3, 6) is one such outcome, the probability of obtaining (3, 6) is 1/36.
The example we just considered consisted of only one outcome of the sample space. If an event consists of only one outcome, it is called a simple event.
We are often interested in finding probabilities of several outcomes represented by an event.
Example (PageIndex{5})
If two dice are rolled, find the probability that the sum of the faces of the dice is 7.
Solution
Let E represent the event that the sum of the faces of two dice is 7.
The possible cases for the sum to be equal to 7 are: (1, 6), (2,5), (3, 4), (4, 3), (5, 2),
and (6, 1), so event E is
E = {(1, 6), (2,5), (3, 4), (4, 3), (5, 2), (6, 1)}
The probability of the event E is
P(E) = 6/36 or 1/6.
Example (PageIndex{6})
One 6 sided die is rolled once. Find the probability that the result is greater than 4.
Solution
The sample space consists of the following six possibilities in set (mathrm{S}): (mathrm{S}={1,2,3,4,5,6})
Let (mathrm{E}) be the event that the number rolled is greater than four: (mathrm{E}={5,6} )
Therefore, the probability of (mathrm{E}) is: (mathrm{P}(mathrm{E}) = 2/6 ext{ or } 1/3).
Example (PageIndex{7})
A jar contains 3 red, 4 white, and 3 blue marbles. If a marble is chosen at random, what is the probability that the marble is a red marble or a blue marble?
Solution
We assume the marbles are (r_1), (r_2), (r_3), (w_1), (w_2), (w_3), (w_4), (b_1), (b_2), (b_3). Let the event (mathrm{C}) represent that the marble is red or blue.
The sample space (mathrm{S}=left{mathrm{r}_{1}, mathrm{r}_{2}, mathrm{r}_{3}, mathrm{w}_{1}, mathrm{w}_{2}, mathrm{w}_{3}, mathrm{w}_{4}, mathrm{b}_{1}, mathrm{b}_{2}, mathrm{b}_{3} ight} ).
And the event (mathrm{C}=left{mathrm{r}_{1}, mathrm{r}_{2}, mathrm{r}_{3}, mathrm{b}_{1}, mathrm{b}_{2}, mathrm{b}_{3} ight})
Therefore, the probability of (mathrm{C}),
[mathrm{P}(mathrm{C})=6 / 10 ext { or } 3 / 5 onumber]
Example (PageIndex{8})
A jar contains three marbles numbered 1, 2, and 3. If two marbles are drawn without replacement, what is the probability that the sum of the numbers is 5?
Note: The two marbles in this example are drawn consecutively without replacement. That means that after a marble is drawn it is not replaced in the jar, and therefore is no longer available to select on the second draw.
Solution
Since two marbles are drawn without replacement, the sample space consists of the following six possibilities.
[mathrm{S}={(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)} onumber]
Note that (1,1), (2,2) and (3,3) are not listed in the sample space. These outcomes are not possible when drawing without replacement, because once the first marble is drawn but not replaced into the jar, that marble is not available in the jar to be selected again on the second draw.
Let the event (mathrm{E}) represent that the sum of the numbers is five. Then
[mathrm{E}={(2,3),(3,2)} onumber]
Therefore, the probability of (mathrm{E}) is
[mathrm{P}(mathrm{E})=2 / 6 ext { or } 1 / 3 onumber. ]
Example (PageIndex{9})
A jar contains three marbles numbered 1, 2, and 3. If two marbles are drawn without replacement, what is the probability that the sum of the numbers is at least 4?
Solution
The sample space, as in Example (PageIndex{7}), consists of the following six possibilities.
[mathrm{S}={(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)} onumber]
Let the event (mathrm{F}) represent that the sum of the numbers is at least four. Then
[mathrm{F}={(1,3),(3,1),(2,3),(3,2)} onumber]
Therefore, the probability of (mathrm{F}) is
[mathrm{P}(mathrm{F})=4 / 6 ext { or } 2 / 3 onumber]
Example (PageIndex{10})
A jar contains three marbles numbered 1, 2, and 3. If two marbles are drawn with replacement, what is the probability that the sum of the numbers is 5?
Note: The two marbles in this example are drawn consecutively with replacement. That means that after a marble is drawn it IS replaced in the jar, and therefore is available to select again on the second draw.
Solution
When two marbles are drawn with replacement, the sample space consists of the following nine possibilities.
[mathrm{S}={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)} onumber]
Note that (1,1), (2,2) and (3,3) are listed in the sample space. These outcomes are possible when drawing with replacement, because once the first marble is drawn and replaced, that marble is not available in the jar to be drawn again.
Let the event E represent that the sum of the numbers is four. Then
[ mathrm{E} = {(2, 3), (3, 2) } onumber]
Therefore, the probability of (mathrm{F}) is (mathrm{P}(mathrm{E}) = 2/9)
Note that in Example (PageIndex{9}) when we selected marbles with replacement, the probability has changed from Example (PageIndex{7}) where we selected marbles without replacement.
Example (PageIndex{11})
A jar contains three marbles numbered 1, 2, and 3. If two marbles are drawn with replacement, what is the probability that the sum of the numbers is at least 4?
Solution
The sample space when drawing with replacement consists of the following nine possibilities.
[ mathrm{S} = {(1,1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3,3)} onumber]
Let the event (mathrm{F}) represent that the sum of the numbers is at least four. Then
[ mathrm{F} = {(1, 3), (3, 1), (2, 3), (3, 2), (2,2), (3,3)} onumber]
Therefore, the probability of (mathrm{F}) is
[mathrm{P}(mathrm{F}) = 6/9 ext{ or } 2/3 onumber.]
Note that in Example (PageIndex{10}) when we selected marbles with replacement, the probability is the same as in Example (PageIndex{8}) where we selected marbles without replacement.
Thus sampling with or without replacement MAY change the probabilities, but may not, depending on the situation in the particular problem under consideration. We’ll re-examine the concepts of sampling with and without replacement in Section 6.3.
Sample Space In Probability
In these lessons, we will learn simple probability, experiments, outcomes, sample space and probability of an event.
The following diagram shows how the sample space for an experiment can be represented by a list, a table, and a tree diagram. Scroll down the page for examples and solutions.
Sample Space
In the study of probability, an experiment is a process or investigation from which results are observed or recorded.
An outcome is a possible result of an experiment.
A sample space is the set of all possible outcomes in the experiment. It is usually denoted by the letter S. Sample space can be written using the set notation, < >.
Experiment 1: Tossing a coin
Possible outcomes are head or tail.
Sample space, S =
Experiment 2: Tossing a die
Possible outcomes are the numbers 1, 2, 3, 4, 5, and 6
Sample space, S =
Experiment 3: Picking a card
In an experiment, a card is picked from a stack of six cards, which spell the word PASCAL.
Possible outcomes are P, A _{1}, S, C, A _{2} and L.
Sample space, S =
Experiment 4: Picking 2 marbles, one at a time, from a bag that contains many blue and red marbles.
Possible outcomes are: (Blue, Blue), (Blue, Red), (Red, Blue) and (Red, Red).
Sample space, S = <(B,B), (B,R), (R,B), (R,R)>.
A simple explanation of Sample Spaces for Probability
Sample Space Of An Event
Sample space is all the possible outcomes of an event. Sometimes the sample space is easy to determine. For example, if you roll a dice, 6 things could happen. You could roll a 1, 2, 3, 4, 5, or 6.
Sometimes sample space is more difficult to determine, so you can make a tree diagram or a list to help you figure out all the possible outcomes.
Example 1:
You are ordering pizza. You can choose a small, medium or large pizza and you can choose cheese or pepperoni. What are the possible ways that you could could order a pizza? How many combinations could you have?
Example 2:
Daisy has 3 pairs of shorts, 2 pairs of shoes and 5 t-shirts. How many outfits can she make?
This lesson is on finding simple probabilities and sample spaces.
Example:
When you roll a die,
Example:
Use the spinner below to answer the following questions:
- What is the sample space?
- P(Blue)
- P(Orange or Green)
- P(Not Red)
- P(Purple)
The following video explains simple probability, experiments, outcomes, sample space and probability of an event. It also gives an example of a simple probability problem.
Example:
A jar contains five balls that are numbered 1 to 5. Also, two of the balls are yellow and the others are red. They are numbered and colored as shown below.
- Find the probability of randomly selecting a red ball.
- Find the probability of randomly selecting an even number ball.
Lists and Sample Spaces - Probability
Example:
Entrees - Ribs, Chicken
Sides - Mac and Cheese, Veggies, Mashed Potatoes
Drinks - Water, Coffee, Milk
What are the different possibilities for the menu?
Explains three methods for listing the sample space of an event and introduces conditional probability: List, Table, Tree Diagram.
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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Course description
This course is an introduction to probability. The mathematical analysis of probabilities originated with attempts to optimize play in various gambling games, and probability continues to be a useful tool for describing many situations in the real world. In this course we will learn the basic ideas of both discrete and continuous probability.
Especially for the latter, the prerequisite for this course is a thorough knowledge of calculus. Math 109, taken at least concurrently, is strongly recommended as we will prove some theorems. The textbook is David F. Anderson, Timo Seppäläinen and Benedek Valkó, Introduction to Probability (Cambridge: Cambridge University Press 2018). We will cover most of chapters 1-6, 8, and 9. Other useful texts include Andrei Nikolaevich Kolmogorov, Foundations of Probability [1], Ani Adhikari and Jim Pitman, Probability for Data Science [2], and Gian-Carlo Rota and Kenneth Baclawski, Introduction to Probability and Random Processes [3].
There will be weekly homework assignments, due in section on Fridays, or before 7:00pm Friday in the appropriate drop box (A01-A04 or A05-A08) in the basement of AP&M. Students are allowed to discuss the homework assignments among themselves, but are expected to turn in their own work &mdash copying someone else's is not acceptable. Homework scores will contribute 15% to the final grade. Occasional extra credit problems are due by specific dates each worth not more than 2% of the final grade.
There will be two midterms, currently planned for Week 4 and Week 8. The final is scheduled for 8:00am Wednesday, March 18. Scores on the two midterms and final will contribute 25%, 25% and 35% to the final grade, respectively * . There will be no makeup tests.
* To be precise, let H_{i}, M1_{i}, M2_{i}, and F_{i} be student i's scores on the homework, on the first and second midterms, and on the final, respectively and let &mu_{H} and &sigma_{H} be the average and standard deviation of the homework scores for the class, respectively. Student i's homework score is rescaled using the average and standard deviation, to get ZH_{i} = (H_{i} - &mu_{H})/&sigma_{H}, with analogous notation for the exam averages, exam standard deviations, and exam Z-scores. Then student i's total Z-score is Z_{i} = 0.15ZH_{i} + 0.25ZM1_{i} + 0.25ZM2_{i} + 0.35ZF_{i}. Any positive Z-scores on extra credit problems will be multiplied by 0.01 each and added to Z_{i}. Student i's letter grade will be assigned based upon max<Z_{i},ZF_{i}>. The cutoffs for grades will be no higher than A-, 0.5 B-, -0.3 C-, -1.3 these cutoffs correspond to about 30% A+,A,A- 30% B+,B,B- and 30% C+,C,C-.
6.1: Sample Spaces and Probability - Mathematics
OBJECTIVE - S.CP.A.1
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (&ldquoor,&rdquo &ldquoand,&rdquo &ldquonot&rdquo).
INTERPRETATION OF OBJECTIVE - S.CP.A.1
This objective has a lot in it for such a short description. This objective lays out the basics of probability by building sample spaces and diagramming them. It then leads into the use of mutually exclusive, complement, intersection and union to describe and diagram certain relationships between those subsets.
(1) The student will be able to determine sample spaces and distinguish between uniform and non-uniform spaces.
(2) The student will be able to diagram and visualize sample spaces using set lists, tree diagrams, tables, and Venn diagrams.
(3) The student will be able to define an outcome as a subset of the sample space and diagram it using a Venn diagram.
(4) The student will be able to use mutually exclusive, complement, intersection and union to describe subsets of sample space and diagram them using a Venn diagram.
Probability is calculated by dividing an outcome (a subset) by the sample space (universal set) and to understand probability you need to understand how subsets work and how they interact with each other. Knowing what mutually exclusive, complement, intersection and union mean will greatly help you visualize and understand many of the probability relationships.
There are definitely a few pitfalls found in this objective. The first is notation. The symbols for union and intersection always seem to be confusing even though I try to tell them that 'U' is for union. Also students struggle to get the concepts of union and intersection when dealing with complements. It just seems that figuring out the union between one subset and not another subset is difficult. Venn diagrams help a lot. Finally and probably the biggest pitfall - Expecting students to know what is in a deck of cards. Kids just don't play cards anymore.
The student needs to come with some basic understanding of probability. Really this is building location - from here we will introduce all of the key foundational ideas of probability.
In that this is the first objective concerning probability and it introduces the foundational concepts and ways to visualize probabilities. everything in this objective connects to the future objectives, especially mutually exclusive, complement, intersection and union.
MY REFLECTIONS (over line l)
I guess my biggest reflection is that I needed to spend more time early one working with the visual representations of complement, intersection and union. When we started dealing with independence and conditional probability a better visual understanding of the subsets would have helped more of my students understand these more difficult idea. Slowdown early and get a real good grasp of Venn diagraming.
Big Ideas Math Book Algebra 2 Answer Key Chapter 10 Probability
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Probability Maintaining Mathematical Proficiency
Write and solve a proportion to answer the question.
Question 1.
What percent of 30 is 6?
Answer:
Question 2.
What number is 68% of 25?
Answer:
Question 3.
34.4 is what percent of 86?
Answer:
Display the data in a histogram.
Question 4.
Answer:
Question 5.
ABSTRACT REASONING
You want to purchase either a sofa or an arm chair at a furniture store. Each item has the same retail price. The sofa is 20% off. The arm chair is 10% off, and you have a coupon to get an additional 10% off the discounted price of the chair. Are the items equally priced after the discounts are applied? Explain.
Answer:
Probability Mathematical Practices
Mathematically proficient students apply the mathematics they know to solve real-life problems.
Monitoring Progress
In Exercises 1 and 2, describe the event as unlikely, equally likely to happen or not happen, or likely. Explain your reasoning.
Question 1.
The oldest child in a family is a girl.
Answer:
Question 2.
The two oldest children in a family with three children are girls.
Answer:
Question 3.
Give an example of an event that is certain to occur.
Answer:
Lesson 10.1 Sample Spaces and Probability
Essential Question How can you list the possible outcomes in the sample space of an experiment?
The sample space of an experiment is the set of all possible outcomes for that experiment.
EXPLORATION 1
Finding the Sample Space of an Experiment
Work with a partner. In an experiment, three coins are flipped. List the possible outcomes in the sample space of the experiment.
EXPLORATION 2
Finding the Sample Space of an Experiment
Work with a partner. List the possible outcomes in the sample space of the experiment.
EXPLORATION 3
Finding the Sample Space of an Experiment
Work with a partner. In an experiment, a spinner is spun.
a. How many ways can you spin a 1? 2? 3? 4? 5?
b. List the sample space.
c. What is the total number of outcomes?
EXPLORATION 4
Finding the Sample Space of an Experiment
Work with a partner. In an experiment, a bag contains 2 blue marbles and 5 red marbles. Two marbles are drawn from the bag.
a. How many ways can you choose two blue? a red then blue? a blue then red? two red?
b. List the sample space.
c. What is the total number of outcomes?
Communicate Your Answer
Question 5.
How can you list the possible outcomes in the sample space of an experiment?
Answer:
Question 6.
For Exploration 3, find the ratio of the number of each possible outcome to the total number of outcomes. Then find the sum of these ratios. Repeat for Exploration 4. What do you observe?
Answer:
Monitoring Progress
Find the number of possible outcomes in the sample space. Then list the possible outcomes.
Question 1.
You flip two coins.
Answer:
Question 2.
You flip two coins and roll a six-sided die.
Answer:
Question 3.
You flip a coin and roll a six-sided die. What is the probability that the coin shows tails and the die shows 4?
Answer:
Question 7.
P(A) = 0.03
Answer:
Question 8.
In Example 4, are you more likely to get 10 points or 5 points?
Answer:
Question 9.
In Example 4, are you more likely to score points (10, 5, or 2) or get 0 points?
Answer:
Question 10.
In Example 5, for which color is the experimental probability of stopping on the color greater than the theoretical probability?
Answer:
Question 11.
In Example 6, what is the probability that a pet-owning adult chosen at random owns a fish?
Answer:
Sample Spaces and Probability 10.1 Exercises
Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
A number that describes the likelihood of an event is the __________ of the event.
Answer:
Question 2.
WRITING
Describe the difference between theoretical probability and experimental probability.
Answer:
Monitoring Progress and Modeling with Mathematics
In Exercises 3–6, find the number of possible outcomes in the sample space. Then list the possible outcomes.
Question 3.
You roll a die and flip three coins.
Answer:
Question 4.
You flip a coin and draw a marble at random from a bag containing two purple marbles and one white marble.
Answer:
Question 5.
A bag contains four red cards numbered 1 through 4, four white cards numbered 1 through 4, and four black cards numbered 1 through 4. You choose a card at random.
Answer:
Question 6.
You draw two marbles without replacement from a bag containing three green marbles and four black marbles.
Answer:
Question 7.
PROBLEM SOLVING
A game show airs on television five days per week. Each day, a prize is randomly placed behind one of two doors. The contestant wins the prize by selecting the correct door. What is the probability that exactly two of the five contestants win a prize during a week?
Answer:
Question 8.
PROBLEM SOLVING
Your friend has two standard decks of 52 playing cards and asks you to randomly draw one card from each deck. What is the probability that you will draw two spades?
Answer:
Question 9.
PROBLEM SOLVING
When two six-sided dice are rolled, there are 36 possible outcomes. Find the probability that (a) the sum is not 4 and (b) the sum is greater than 5.
Answer:
Question 10.
PROBLEM SOLVING
The age distribution of a population is shown. Find the probability of each event.
a. A person chosen at random is at least 15 years old.
b. A person chosen at random is from 25 to 44 years old.
Answer:
Question 11.
ERROR ANALYSIS
A student randomly guesses the answers to two true-false questions. Describe and correct the error in finding the probability of the student guessing both answers correctly.
Answer:
Question 12.
ERROR ANALYSIS
A student randomly draws a number between 1 and 30. Describe and correct the error in finding the probability that the number drawn is greater than 4.
Answer:
Question 13.
MATHEMATICAL CONNECTIONS
You throw a dart at the board shown. Your dart is equally likely to hit any point inside the square board. What is the probability your dart lands in the yellow region?
Answer:
Question 14.
MATHEMATICAL CONNECTIONS
The map shows the length (in miles) of shoreline along the Gulf of Mexico for each state that borders the body of water. What is the probability that a ship coming ashore at a random point in the Gulf of Mexico lands in the given state?
a. Texas
b. Alabama
c. Florida
d. Louisiana
Answer:
Question 15.
DRAWING CONCLUSIONS
You roll a six-sided die 60 times. The table shows the results. For which number is the experimental probability of rolling the number the same as the theoretical probability?
Answer:
Question 16.
DRAWING CONCLUSIONS
A bag contains 5 marbles that are each a different color. A marble is drawn, its color is recorded, and then the marble is placed back in the bag. This process is repeated until 30 marbles have been drawn. The table shows the results. For which marble is the experimental probability of drawing the marble the same as the theoretical probability?
Answer:
Question 17.
REASONING
Refer to the spinner shown. The spinner is divided into sections with the same area.
a. What is the theoretical probability that the spinner stops on a multiple of 3?
b. You spin the spinner 30 times. It stops on a multiple of 3 twenty times. What is the experimental probability of stopping on a multiple of 3?
c. Explain why the probability you found in part (b) is different than the probability you found in part (a).
Answer:
Question 18.
OPEN-ENDED
Describe a real-life event that has a probability of 0. Then describe a real-life event that has a probability of 1.
Answer:
Question 19.
DRAWING CONCLUSIONS
A survey of 2237 adults ages 18 and over asked which sport is their favorite. The results are shown in the figure. What is the probability that an adult chosen at random prefers auto racing?
Answer:
Question 20.
DRAWING CONCLUSIONS
A survey of 2392 adults ages 18 and over asked what type of food they would be most likely to choose at a restaurant. The results are shown in the figure. What is the probability that an adult chosen at random prefers Italian food?
Answer:
Question 21.
ANALYZING RELATIONSHIPS
Refer to the board in Exercise 13. Order the likelihoods that the dart lands in the given region from least likely to most likely.
A. green
B. not blue
C. red
D. not yellow
Answer:
Question 22.
ANALYZING RELATIONSHIPS
Refer to the chart below. Order the following events from least likely to most likely.
A. It rains on Sunday.
B. It does not rain on Saturday.
C. It rains on Monday.
D. It does not rain on Friday.
Answer:
Question 23.
USING TOOLS
Use the figure in Example 3 to answer each question.
a. List the possible sums that result from rolling two six-sided dice.
b. Find the theoretical probability of rolling each sum.
c. The table below shows a simulation of rolling two six-sided dice three times. Use a random number generator to simulate rolling two six-sided dice 50 times. Compare the experimental probabilities of rolling each sum with the theoretical probabilities.
Answer:
Question 24.
MAKING AN ARGUMENT
You flip a coin three times. It lands on heads twice and on tails once. Your friend concludes that the theoretical probability of the coin landing heads up is P(heads up) = (frac<2><3>). Is your friend correct? Explain your reasoning.
Answer:
Question 25.
MATHEMATICAL CONNECTIONS
A sphere fits inside a cube so that it touches each side, as shown. What is the probability a point chosen at random inside the cube is also inside the sphere?
Answer:
Question 26.
HOW DO YOU SEE IT?
Consider the graph of f shown. What is the probability that the graph of y = f (x) +c intersects the x-axis when c is a randomly chosen integer from 1 to 6? Explain.
Answer:
Question 27.
DRAWING CONCLUSIONS
A manufacturer tests 1200 computers and finds that 9 of them have defects. Find the probability that a computer chosen at random has a defect. Predict the number of computers with defects in a shipment of 15,000 computers. Explain your reasoning.
Answer:
Question 28.
THOUGHT PROVOKING
The tree diagram shows a sample space. Write a probability problem that can be represented by the sample space. Then write the answer(s) to the problem.
Answer:
Maintaining Mathematical Proficiency
Find the product or quotient.
Question 29.
(frac<3 x>
Answer:
Question 31.
(frac
Answer:
Question 33.
(frac<3 x> <12 x-11>div frac
Answer:
Question 34.
(frac<3 x^<2>+2 x-13>
Answer:
Lesson 10.2 Independent and Dependent Events
Essential Question How can you determine whether two events are independent or dependent?
Two events are independent events when the occurrence of one event does not affect the occurrence of the other event. Two events are dependent events when the occurrence of one event does affect the occurrence of the other event.
EXPLORATION 1
Identifying Independent and Dependent Events
Work with a partner. Determine whether the events are independent or dependent. Explain your reasoning.
a. Two six-sided dice are rolled.
b. Six pieces of paper, numbered 1 through 6, are in a bag. Two pieces of paper are selected one at a time without replacement.
EXPLORATION 2
Finding Experimental Probabilities
Work with a partner.
a. In Exploration 1(a), experimentally estimate the probability that the sum of the two numbers rolled is 7. Describe your experiment.
b. In Exploration 1(b), experimentally estimate the probability that the sum of the two numbers selected is 7. Describe your experiment.
EXPLORATION 3
Finding Theoretical Probabilities
Work with a partner.
a. In Exploration 1(a), find the theoretical probability that the sum of the two numbers rolled is 7. Then compare your answer with the experimental probability you found in Exploration 2(a).
b. In Exploration 1(b), find the theoretical probability that the sum of the two numbers selected is 7. Then compare your answer with the experimental probability you found in Exploration 2(b).
c. Compare the probabilities you obtained in parts (a) and (b).
Communicate Your Answer
Question 4.
How can you determine whether two events are independent or dependent?
Answer:
Question 5.
Determine whether the events are independent or dependent. Explain your reasoning.
a. You roll a 4 on a six-sided die and spin red on a spinner.
b. Your teacher chooses a student to lead a group, chooses another student to lead a second group, and chooses a third student to lead a third group.
Answer:
Monitoring Progress
Question 1.
In Example 1, determine whether guessing Question 1 incorrectly and guessing Question 2 correctly are independent events.
Answer:
Question 2.
In Example 2, determine whether randomly selecting a girl first and randomly selecting a boy second are independent events.
Answer:
Question 3.
In Example 3, what is the probability that you spin an even number and then an odd number?
Answer:
Question 4.
In Example 4, what is the probability that both bills are $1 bills?
Answer:
Question 5.
In Example 5, what is the probability that none of the cards drawn are hearts when (a) you replace each card, and (b) you do not replace each card? Compare the probabilities.
Answer:
Question 6.
In Example 6, find (a) the probability that a non-defective part “passes,” and (b) the probability that a defective part “fails.”
Answer:
Question 7.
At a coffee shop, 80% of customers order coffee. Only 15% of customers order coffee and a bagel. What is the probability that a customer who orders coffee also orders a bagel?
Answer:
Independent and Dependent Events 10.2 Exercises
Vocabulary and Core Concept Check
Question 1.
WRITING
Explain the difference between dependent events and independent events, and give an example of each.
Answer:
Question 2.
COMPLETE THE SENTENCE
The probability that event B will occur given that event A has occurred is called the __________ of B given A and is written as _________
Answer:
Monitoring Progress and Modeling with Mathematics
In Exercises 3–6, tell whether the events are independent or dependent. Explain your reasoning.
Question 3.
A box of granola bars contains an assortment of flavors. You randomly choose a granola bar and eat it. Then you randomly choose another bar.
Event A: You choose a coconut almond bar first.
Event B: You choose a cranberry almond bar second.
Answer:
Question 4.
You roll a six-sided die and flip a coin.
Event A: You get a 4 when rolling the die.
Event B: You get tails when flipping the coin.
Answer:
Question 5.
Your MP3 player contains hip-hop and rock songs. You randomly choose a song. Then you randomly choose another song without repeating song choices.
Event A: You choose a hip-hop song first.
Event B: You choose a rock song second.
Answer:
Question 6.
There are 22 novels of various genres on a shelf. You randomly choose a novel and put it back. Then you randomly choose another novel.
Event A: You choose a mystery novel.
Event B: You choose a science fiction novel.
Answer:
In Exercises 7–10, determine whether the events are independent.
Question 7.
You play a game that involves spinning a wheel. Each section of the wheel shown has the same area. Use a sample space to determine whether randomly spinning blue and then green are independent events.
Answer:
Question 8.
You have one red apple and three green apples in a bowl. You randomly select one apple to eat now and another apple for your lunch. Use a sample space to determine whether randomly selecting a green apple first and randomly selecting a green apple second are independent events.
Answer:
Question 9.
A student is taking a multiple-choice test where each question has four choices. The student randomly guesses the answers to the five-question test. Use a sample space to determine whether guessing Question 1 correctly and Question 2 correctly are independent events.
Answer:
Question 10.
A vase contains four white roses and one red rose. You randomly select two roses to take home. Use a sample space to determine whether randomly selecting a white rose first and randomly selecting a white rose second are independent events.
Answer:
Question 11.
PROBLEM SOLVING
You play a game that involves spinning the money wheel shown. You spin the wheel twice. Find the probability that you get more than $500 on your first spin and then go bankrupt on your second spin.
Answer:
Question 12.
PROBLEM SOLVING
You play a game that involves drawing two numbers from a hat. There are 25 pieces of paper numbered from 1 to 25 in the hat. Each number is replaced after it is drawn. Find the probability that you will draw the 3 on your first draw and a number greater than 10 on your second draw.
Answer:
Question 13.
PROBLEM SOLVING
A drawer contains 12 white socks and 8 black socks. You randomly choose 1 sock and do not replace it. Then you randomly choose another sock. Find the probability that both events A and B will occur.
Event A: The first sock is white.
Event B: The second sock is white.
Answer:
Question 14.
PROBLEM SOLVING
A word game has 100 tiles, 98 of which are letters and 2 of which are blank. The numbers of tiles of each letter are shown. Yourandomly draw 1 tile, set it aside, and then randomly draw another tile. Find the probability that both events A and B will occur.
Answer:
Question 15.
ERROR ANALYSIS
Events A and B are independent. Describe and correct the error in finding P(A and B).
Answer:
Question 16.
ERROR ANALYSIS
A shelf contains 3 fashion magazines and 4 health magazines. You randomly choose one to read, set it aside, and randomly choose another for your friend to read. Describe and correct the error in finding the probability that both events A and B occur.
Event A: The first magazine is fashion.
Event B: The second magazine is health.
Answer:
Question 17.
NUMBER SENSE
Events A and B are independent. Suppose P(B) = 0.4 and P(A and B) = 0.13. Find P(A).
Answer:
Question 18.
NUMBER SENSE
Events A and B are dependent. Suppose P(B | A) = 0.6 and P(A and B) = 0.15. Find P(A).
Answer:
Question 19.
ANALYZING RELATIONSHIPS
You randomly select three cards from a standard deck of 52 playing cards. What is the probability that all three cards are face cards when (a) you replace each card before selecting the next card, and (b) you do not replace each card before selecting the next card? Compare the probabilities.
Answer:
Question 20.
ANALYZING RELATIONSHIPS
A bag contains 9 red marbles, 4 blue marbles, and 7 yellow marbles. You randomly select three marbles from the bag. What is the probability that all three marbles are red when (a) you replace each marble before selecting the next marble, and (b) you do not replace each marble before selecting the next marble? Compare the probabilities.
Answer:
Question 21.
ATTEND TO PRECISION
The table shows the number of species in the United States listed as endangered and threatened. Find (a) the probability that a randomly selected endangered species is a bird, and (b) the probability that a randomly selected mammal is endangered.
Answer:
Question 22.
ATTEND TO PRECISION
The table shows the number of tropical cyclones that formed during the hurricane seasons over a 12-year period. Find (a) the probability to predict whether a future tropical cyclone in the Northern Hemisphere is a hurricane, and (b) the probability to predict whether a hurricane is in the Southern Hemisphere.
Answer:
Question 23.
PROBLEM SOLVING
At a school, 43% of students attend the homecoming football game. Only 23% of students go to the game and the homecoming dance. What is the probability that a student who attends thefootball game also attends the dance?
Answer:
Question 24.
PROBLEM SOLVING
At a gas station, 84% of customers buy gasoline. Only 5% of customers buy gasoline and a beverage. What is the probability that a customer who buys gasoline also buys a beverage?
Answer:
Question 25.
PROBLEM SOLVING
You and 19 other students volunteer to present the “Best Teacher” award at a school banquet. One student volunteer will be chosen to present the award. Each student worked at least 1 hour in preparation for the banquet. You worked for 4 hours, and the group worked a combined total of 45 hours. For each situation, describe a process that gives you a “fair” chance to be chosen, and find the probability that you are chosen.
a. “Fair” means equally likely.
b. “Fair” means proportional to the number of hours each student worked in preparation.
Answer:
Question 26.
HOW DO YOU SEE IT?
A bag contains one red marble and one blue marble. The diagrams show the possible outcomes of randomly choosing two marbles using different methods. For each method, determine whether the marbles were selected with or without replacement.
Answer:
Question 27.
MAKING AN ARGUMENT
A meteorologist claims that there is a 70% chance of rain. When it rains, there is a 75% chance that your softball game will be rescheduled. Your friend believes the game is more likely to be rescheduled than played. Is your friend correct? Explain your reasoning.
Answer:
Question 28.
THOUGHT PROVOKING
Two six-sided dice are rolled once. Events A and B are represented by the diagram. Describe each event. Are the two events dependent or independent? Justify your reasoning.
Answer:
Question 29.
MODELING WITH MATHEMATICS
A football team is losing by 14 points near the end of a game. The team scores two touchdowns (worth 6 points each) before the end of the game. After each touchdown, the coach must decide whether to go for 1 point with a kick (which is successful 99% of the time) or 2 points with a run or pass (which is successful 45% of the time).
a. If the team goes for 1 point after each touchdown, what is the probability that the team wins? loses? ties?
b. If the team goes for 2 points after each touchdown, what is the probability that the team wins? loses? ties?
c. Can you develop a strategy so that the coach’s team has a probability of winning the game that is greater than the probability of losing? If so, explain your strategy and calculate the probabilities of winning and losing the game.
Answer:
Question 30.
ABSTRACT REASONING
Assume that A and B are independent events.
a. Explain why P(B) = P(B|A) and P(A) =P(A|B).
b. Can P(A and B) also be defined as P(B) • P(A|B)? Justify your reasoning.
Answer:
Maintaining Mathematical Proficiency
Solve the equation. Check your solution.
Question 31.
(frac<9><10>)x = 0.18
Answer:
Question 32.
(frac<1><4>)x + 0.5x = 1.5
Answer:
Question 33.
0.3x − (frac<3><5>)x + 1.6 = 1.555
Answer:
Lesson 10.3 Two-Way Tables and Probability
Essential Question How can you construct and interpret a two-way table?
EXPLORATION 1
Completing and Using a Two-Way Table
Work with a partner. A two-way table displays the same information as a Venn diagram. In a two-way table, one category is represented by the rows and the other category is represented by the columns.
The Venn diagram shows the results of a survey in which 80 students were asked whether they play a musical instrument and whether they speak a foreign language. Use the Venn diagram to complete the two-way table. Then use the two-way table to answer each question.
a. How many students play an instrument?
b. How many students speak a foreign language?
c. How many students play an instrument and speak a foreign language?
d. How many students do not play an instrument and do not speak a foreign language?
e. How many students play an instrument and do not speak a foreign language?
EXPLORATION 2
Two-Way Tables and Probability
Work with a partner. In Exploration 1, one student is selected at random from the 80 students who took the survey. Find the probability that the student
a. plays an instrument.
b. speaks a foreign language.
c. plays an instrument and speaks a foreign language.
d. does not play an instrument and does not speak a foreign language.
e. plays an instrument and does not speak a foreign language.
EXPLORATION 3
Conducting a Survey
Work with your class. Conduct a survey of the students in your class. Choose two categories that are different from those given in Explorations 1 and 2. Then summarize the results in both a Venn diagram and a two-way table. Discuss the results.
Communicate Your Answer
Question 4.
How can you construct and interpret a two-way table?
Answer:
Question 5.
How can you use a two-way table to determine probabilities?
Answer:
Monitoring Progress
Question 1.
You randomly survey students about whether they are in favor of planting a community garden at school. Of 96 boys surveyed, 61 are in favor. Of 88 girls surveyed, 17 are against. Organize the results in a two-way table. Then find and interpret the marginal frequencies.
Answer:
Question 2.
Use the survey results in Monitoring Progress Question 1 to make a two-way table that shows the joint and marginal relative frequencies.
Answer:
Question 3.
Use the survey results in Example 1 to make a two-way table that shows the conditional relative frequencies based on the column totals. Interpret the conditional relative frequencies in the context of the problem.
Answer:
Question 4.
Use the survey results in Monitoring Progress Question 1 to make a two-way table that shows the conditional relative frequencies based on the row totals. Interpret the conditional relative frequencies in the context of the problem.
Answer:
Question 5.
In Example 4, what is the probability that a randomly selected customer who is located in Santa Monica will not recommend the provider to a friend?
Answer:
Question 6.
In Example 4, determine whether recommending the provider to a friend and living in Santa Monica are independent events. Explain your reasoning.
Answer:
Question 7.
A manager is assessing three employees in order to offer one of them a promotion. Over a period of time, the manager records whether the employees meet or exceed expectations on their assigned tasks. The table shows the manager’s results. Which employee should be offered the promotion? Explain.
Answer:
Two-Way Tables and Probability 10.3 Exercises
Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
A(n) _____________ displays data collected from the same source that belongs to two different categories.
Answer:
Question 2.
WRITING
Compare the definitions of joint relative frequency, marginal relative frequency, and conditional relative frequency.
Answer:
Monitoring Progress and Modeling with Mathematics
In Exercises 3 and 4, complete the two-way table.
Question 3.
Answer:
Question 4.
Answer:
Question 5.
MODELING WITH MATHEMATICS
You survey 171 males and 180 females at Grand Central Station in New York City. Of those, 132 males and 151 females wash their hands after using the public rest rooms. Organize these results in a two-way table. Then find and interpret the marginal frequencies.
Answer:
Question 6.
MODELING WITH MATHEMATICS
A survey asks 60 teachers and 48 parents whether school uniforms reduce distractions in school. Of those, 49 teachers and 18 parents say uniforms reduce distractions in school. Organize these results in a two-way table. Then find and interpret the marginal frequencies.
Answer:
USING STRUCTURE In Exercises 7 and 8, use the two-way table to create a two-way table that shows the joint and marginal relative frequencies.
Question 7.
Answer:
Question 8.
Answer:
Question 9.
MODELING WITH MATHEMATICS
Use the survey results from Exercise 5 to make a two-way table that shows the joint and marginal relative frequencies.
Answer:
Question 10.
MODELING WITH MATHEMATICS
In a survey, 49 people received a flu vaccine before the flu season and 63 people did not receive the vaccine. Of those who receive the flu vaccine, 16 people got the flu. Of those who did not receive the vaccine, 17 got the flu. Make a two-way table that shows the joint and marginal relative frequencies.
Answer:
Question 11.
MODELING WITH MATHEMATICS
A survey finds that 110 people ate breakfast and 30 people skipped breakfast. Of those who ate breakfast, 10 people felt tired. Of those who skipped breakfast, 10 people felt tired. Make a two-way table that shows the conditional relative frequencies based on the breakfast totals.
Answer:
Question 12.
MODELING WITH MATHEMATICS
Use the survey results from Exercise 10 to make a two-way table that shows the conditional relative frequencies based on the flu vaccine totals.
Answer:
Question 13.
PROBLEM SOLVING
Three different local hospitals in New York surveyed their patients. The survey asked whether the patient’s physician communicated efficiently. The results, given as joint relative frequencies, are shown in the two-way table.
a. What is the probability that a randomly selected patient located in Saratoga was satisfied with the communication of the physician?
b. What is the probability that a randomly selected patient who was not satisfied with the physician’s communication is located in Glens Falls?
c. Determine whether being satisfied with the communication of the physician and living in Saratoga are independent events.
Answer:
Question 14.
PROBLEM SOLVING
A researcher surveys a random sample of high school students in seven states. The survey asks whether students plan to stay in their home state after graduation. The results, given as joint relative frequencies, are shown in the two-way table.
a. What is the probability that a randomly selected student who lives in Nebraska plans to stay in his or her home state after graduation?
b. What is the probability that a randomly selected student who does not plan to stay in his or her home state after graduation lives in North Carolina?
c. Determine whether planning to stay in their home state and living in Nebraska are independent events.
Answer:
ERROR ANALYSIS In Exercises 15 and 16, describe and correct the error in finding the given conditional probability.
Question 15.
P(yes|Tokyo)
Answer:
Question 16.
P(London|no)
Answer:
Question 17.
PROBLEM SOLVING
You want to find the quickest route to school. You map out three routes. Before school, you randomly select a route and record whether you are late or on time. The table shows your findings. Assuming you leave at the same time each morning, which route should you use? Explain.
Answer:
Question 18.
PROBLEM SOLVING
A teacher is assessing three groups of students in order to offer one group a prize. Over a period of time, the teacher records whether the groups meet or exceed expectations on their assigned tasks. The table shows the teacher’s results. Which group should be awarded the prize? Explain.
Answer:
Question 19.
OPEN-ENDED
Create and conduct a survey in your class. Organize the results in a two-way table. Then create a two-way table that shows the joint and marginal frequencies.
Answer:
Question 20.
HOW DO YOU SEE IT?
A research group surveys parents and coaches of high school students about whether competitive sports are important in school. The two-way table shows the results of the survey.
a. What does 120 represent?
b. What does 1336 represent?
c. What does 1501 represent?
Answer:
Question 21.
MAKING AN ARGUMENT
Your friend uses the table below to determine which workout routine is the best. Your friend decides that Routine B is the best option because it has the fewest tally marks in the “Does Not Reach Goal” column. Is your friend correct? Explain your reasoning.
Answer:
Question 22.
MODELING WITH MATHEMATICS
A survey asks students whether they prefer math class or science class. Of the 150 male students surveyed, 62% prefer math class over science class. Of the female students surveyed,74% prefer math. Construct a two-way table to show the number of students in each category if 350 students were surveyed.
Answer:
Question 23.
MULTIPLE REPRESENTATIONS
Use the Venn diagram to construct a two-way table. Then use your table to answer the questions.
a.What is the probability that a randomly selected person does not own either pet?
b. What is the probability that a randomly selected person who owns a dog also owns a cat?
Answer:
Question 24.
WRITING
Compare two-way tables and Venn diagrams. Then describe the advantages and disadvantages of each.
Answer:
Question 25.
PROBLEM SOLVING
A company creates a new snack, N,and tests it against its current leader, L. The table shows the results.
The company is deciding whether it should try to improve the snack before marketing it, and to whom the snack should be marketed. Use probability to explain the decisions the company should make when the total size of the snack’s market is expected to (a) change very little, and (b) expand very rapidly.
Answer:
Question 26.
THOUGHT PROVOKING
Bayes’ Theorem is given by
P(A|B) = (frac
).
Use a two-way table to write an example of Bayes’ Theorem.
Answer:
Maintaining Mathematical Proficiency
Draw a Venn diagram of the sets described.
Question 27.
Of the positive integers less than 15, set A consists of the factors of 15 and set B consists of all odd numbers.
Answer:
Question 28.
Of the positive integers less than 14, set A consists of all prime numbers and set B consists of all even numbers.
Answer:
Question 29.
Of the positive integers less than 24, set A consists of the multiples of 2 and set B consists of all the multiples of 3.
Answer:
Probability Study Skills: Making a Mental Cheat Sheet
10.1–10.3 What Did You Learn?
Core Vocabulary
Core Concepts
Section 10.1
Theoretical Probabilities, p. 538
Probability of the Complement of an Event, p. 539
Experimental Probabilities, p. 541
Section 10.2
Probability of Independent Events, p. 546
Probability of Dependent Events, p. 547
Finding Conditional Probabilities, p. 549
Section 10.3
Making Two-Way Tables, p. 554
Relative and Conditional Relative Frequencies, p. 555
Mathematical Practices
Question 1.
How can you use a number line to analyze the error in Exercise 12 on page 542?
Answer:
Question 2.
Explain how you used probability to correct the flawed logic of your friend in Exercise 21 on page 560.
Answer:
Study Skills: Making a Mental Cheat Sheet
- Write down important information on note cards.
- Memorize the information on the note cards, placing the ones containing information you know in one stack and the ones containing information you do not know in another stack. Keep working on the information you do not know.
Probability 10.1–10.3 Quiz
Question 1.
You randomly draw a marble out of a bag containing 8 green marbles, 4 blue marbles, 12 yellow marbles, and 10 red marbles. Find the probability of drawing a marble that is not yellow.
Answer:
Question 4.
P(A) = 0.01
Answer:
Question 5.
You roll a six-sided die 30 times. A 5 is rolled 8 times. What is the theoretical probability of rolling a 5? What is the experimental probability of rolling a 5?
Answer:
Question 6.
Events A and B are independent. Find the missing probability.
P(A) = 0.25
P(B) = _____
P(A and B) = 0.05
Answer:
Question 7.
Events A and B are dependent. Find the missing probability.
P(A) = 0.6
P(B|A) = 0.2
P(A and B) = _____
Answer:
Question 8.
Find the probability that a dart thrown at the circular target shown will hit the given region. Assume the dart is equally likely to hit any point inside the target.
a. the center circle
b. outside the square
c. inside the square but outside the center circle
Answer:
Question 9.
A survey asks 13-year-old and 15-year-old students about their eating habits. Four hundred students are surveyed, 100 male students and 100 female students from each age group. The bar graph shows the number of students who said they eat fruit every day.
a. Find the probability that a female student, chosen at random from the students surveyed, eats fruit every day.
b. Find the probability that a 15-year-old student, chosen at random from the students surveyed, eats fruit every day.
Answer:
Question 10.
There are 14 boys and 18 girls in a class. The teacher allows the students to vote whether they want to take a test on Friday or on Monday. A total of 6 boys and 10 girls vote to take the test on Friday. Organize the information in a two-way table. Then find and interpret the marginal frequencies.
Answer:
Question 11.
Three schools compete in a cross country invitational. Of the 15 athletes on your team, 9 achieve their goal times. Of the 20 athletes on the home team, 6 achieve their goal times. On your rival’s team, 8 of the 13 athletes achieve their goal times. Organize the information in a two-way table. Then determine the probability that a randomly selected runner who achieves his or her goal time is from your school.
Answer:
Lesson 10.4 Probability of Disjoint and Overlapping Events
Essential Question How can you find probabilities of disjoint and overlapping events?
Two events are disjoint, or mutually exclusive, when they have no outcomes in common. Two events are overlapping when they have one or more outcomes in common.
EXPLORATION 1
Disjoint Events and Overlapping Events
Work with a partner. A six-sided die is rolled. Draw a Venn diagram that relates the two events. Then decide whether the events are disjoint or overlapping.
a. Event A: The result is an even number.
Event B: The result is a prime number.
b. Event A: The result is 2 or 4.
Event B: The result is an odd number.
EXPLORATION 2
Finding the Probability that Two Events Occur
Work with a partner. A six-sided die is rolled. For each pair of events, find (a) P(A), (b) P(B), (c) P(A and B), and (d) P(A or B).
a. Event A: The result is an even number.
Event B: The result is a prime number.
b. Event A: The result is 2 or 4.
Event B: The result is an odd number.
EXPLORATION 3
Discovering Probability Formulas
Work with a partner.
a. In general, if event A and event B are disjoint, then what is the probability that event A or event B will occur? Use a Venn diagram to justify your conclusion.
b. In general, if event A and event B are overlapping, then what is the probability that event A or event B will occur? Use a Venn diagram to justify your conclusion.
c. Conduct an experiment using a six-sided die. Roll the die 50 times and record the results. Then use the results to find the probabilities described in Exploration 2. How closely do your experimental probabilities compare to the theoretical probabilities you found in Exploration 2?
Communicate Your Answer
Question 4.
How can you find probabilities of disjoint and overlapping events?
Answer:
Question 5.
Give examples of disjoint events and overlapping events that do not involve dice.
Answer:
Monitoring Progress
A card is randomly selected from a standard deck of 52 playing cards. Find the probability of the event.
Question 1.
selecting an ace or an 8
Answer:
Question 2.
selecting a 10 or a diamond
Answer:
Question 3.
WHAT IF?
In Example 3, suppose 32 seniors are in the band and 64 seniors are in the band or on the honor roll. What is the probability that a randomly selected senior is both in the band and on the honor roll?
Answer:
Question 4.
In Example 4, what is the probability that the diagnosis is incorrect?
Answer:
Question 5.
A high school basketball team leads at halftime in 60% of the games in a season. The team wins 80% of the time when they have the halftime lead, but only 10% of the time when they do not. What is the probability that the team wins a particular game during the season?
Answer:
Probability of Disjoint and Overlapping Events 10.4 Exercises
Vocabulary and Core Concept Check
Question 1.
WRITING
Are the events A and (ar) disjoint? Explain. Then give an example of a real-life event and its complement.
Answer:
Question 2.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.
Answer:
Monitoring Progress and Modeling with Mathematics
In Exercises 3–6, events A and B are disjoint. Find P(A or B).
Question 3.
P(A) = 0.3, P(B) = 0.1
Answer:
Question 4.
P(A) = 0.55, P(B) = 0.2
Answer:
Question 5.
P(A) = (frac<1><3>), P(B) = (frac<1><4>)
Answer:
Question 7.
PROBLEM SOLVING
Your dart is equally likely to hit any point inside the board shown. You throw a dart and pop a balloon. What is the probability that the balloon is red or blue?
Answer:
Question 8.
PROBLEM SOLVING
You and your friend are among several candidates running for class president. You estimate that there is a 45% chance you will win and a 25% chance your friend will win. What is the probability that you or your friend win the election?
Answer:
Question 9.
PROBLEM SOLVING
You are performing an experiment to determine how well plants grow under different light sources. Of the 30 plants in the experiment, 12 receive visible light, 15 receive ultraviolet light, and 6 receive both visible and ultraviolet light. What is the probability that a plant in the experiment receives visible or ultraviolet light?
Answer:
Question 10.
PROBLEM SOLVING
Of 162 students honored at an academic awards banquet, 48 won awards for mathematics and 78 won awards for English. There are 14 students who won awards for both mathematics and English. A newspaper chooses a student at random for an interview. What is the probability that the student interviewed won an award for English or mathematics?
Answer:
ERROR ANALYSIS In Exercises 11 and 12, describe and correct the error in finding the probability of randomly drawing the given card from a standard deck of 52 playing cards.
Question 11.
Answer:
Question 12.
Answer:
In Exercises 13 and 14, you roll a six-sided die. Find P(A or B).
Question 13.
Event A: Roll a 6.
Event B: Roll a prime number.
Answer:
Question 14.
Event A: Roll an odd number.
Event B: Roll a number less than 5.
Answer:
Question 15.
DRAWING CONCLUSIONS
A group of 40 trees in a forest are not growing properly. A botanist determines that 34 of the trees have a disease or are being damaged by insects, with 18 trees having a disease and 20 being damaged by insects. What is the probability that a randomly selected tree has both a disease and is being damaged by insects?
Answer:
Question 16.
DRAWING CONCLUSIONS
A company paid overtime wages or hired temporary help during 9 months of the year. Overtime wages were paid during 7 months, and temporary help was hired during 4 months. At the end of the year, an auditor examines the accounting records and randomly selects one month to check the payroll. What is the probability that the auditor will select a month in which the company paid overtime wages and hired temporary help?
Answer:
Question 17.
DRAWING CONCLUSIONS
A company is focus testing a new type of fruit drink. The focus group is 47% male. Of the responses, 40% of the males and 54% of the females said they would buy the fruit drink. What is the probability that a randomly selected person would buy the fruit drink?
Answer:
Question 18.
DRAWING CONCLUSIONS
The Redbirds trail the Bluebirds by one goal with 1 minute left in the hockey game. The Redbirds’ coach must decide whether to remove the goalie and add a frontline player. The probabilities of each team scoring are shown in the table.
a. Find the probability that the Redbirds score and the Bluebirds do not score when the coach leaves the goalie in.
b. Find the probability that the Redbirds score and the Bluebirds do not score when the coach takes the goalie out.
c. Based on parts (a) and (b), what should the coach do?
Answer:
Question 19.
PROBLEM SOLVING
You can win concert tickets from a radio station if you are the first person to call when the song of the day is played, or if you are the first person to correctly answer the trivia question. The song of the day is announced at a random time between 7:00 and 7:30 A.M. The trivia question is asked at a random time between 7:15 and 7:45 A.M. You begin listening to the radio station at 7:20. Find the probability that you miss the announcement of the song of the day or the trivia question.
Answer:
Question 20.
HOW DO YOU SEE IT?
Are events A and B disjoint events? Explain your reasoning.
Answer:
Question 21.
PROBLEM SOLVING
You take a bus from your neighborhood to your school. The express bus arrives at your neighborhood at a random time between 7:30 and 7:36 A.M. The local bus arrives at your neighborhood at a random time between 7:30 and 7:40 A.M. You arrive at the bus stop at 7:33 A.M. Find the probability that you missed both the express bus and the local bus.
Answer:
Question 22.
THOUGHT PROVOKING
Write a general rule for finding P(A or B or C) for (a) disjoint and (b) overlapping events A, B, and C.
Answer:
Question 23.
MAKING AN ARGUMENT
A bag contains 40 cards numbered 1 through 40 that are either red or blue. A card is drawn at random and placed back in the bag. This is done four times. Two red cards are drawn, numbered 31 and 19, and two blue cards are drawn, numbered 22 and 7. Your friend concludes that red cards and even numbers must be mutually exclusive. Is your friend correct? Explain.
Answer:
Maintaining Mathematical Proficiency
Write the first six terms of the sequence.
Question 24.
a_{1} = 4, a_{n} = 2a_{n-1} + 3
Answer:
Question 25.
a_{1} = 1, a_{n} = (frac
Answer:
Lesson 10.5 Permutations and Combinations
Essential Question How can a tree diagram help you visualize the number of ways in which two or more events can occur?
EXPLORATION 1
Reading a Tree Diagram
Work with a partner. Two coins are flipped and the spinner is spun. The tree diagram shows the possible outcomes.
a. How many outcomes are possible?
b. List the possible outcomes.
EXPLORATION 2
Reading a Tree Diagram
Work with a partner. Consider the tree diagram below.
a. How many events are shown?
b. What outcomes are possible for each event?
c. How many outcomes are possible?
d. List the possible outcomes.
EXPLORATION 3
Writing a Conjecture
Work with a partner.
a. Consider the following general problem: Event 1 can occur in m ways and event 2 can occur in n ways. Write a conjecture about the number of ways the two events can occur. Explain your reasoning.
b. Use the conjecture you wrote in part (a) to write a conjecture about the number of ways more than two events can occur. Explain your reasoning.
c. Use the results of Explorations 1(a) and 2(c) to verify your conjectures.
Communicate Your Answer
Question 4.
How can a tree diagram help you visualize the number of ways in which two or more events can occur?
Answer:
Question 5.
In Exploration 1, the spinner is spun a second time. How many outcomes are possible?
Answer:
Monitoring Progress
Question 1.
In how many ways can you arrange the letters in the word HOUSE?
Answer:
Question 2.
In how many ways can you arrange 3 of the letters in the word MARCH?
Answer:
Question 3.
WHAT IF?
In Example 2, suppose there are 8 horses in the race. In how many different ways can the horses finish first, second, and third? (Assume there are no ties.)
Answer:
Question 4.
WHAT IF?
In Example 3, suppose there are 14 floats in the parade. Find the probability that the soccer team is first and the chorus is second.
Answer:
Question 5.
Count the possible combinations of 3 letters chosen from the list A, B, C, D, E.
Answer:
Question 6.
WHAT IF?
In Example 5, suppose you can choose 3 side dishes out of the list of 8 side dishes. How many combinations are possible?
Answer:
Question 7.
WHAT IF?
In Example 6, suppose there are 20 photos in the collage. Find the probability that your photo and your friend’s photo are the 2 placed at the top of the page.
Answer:
Question 8.
Use the Binomial Theorem to write the expansion of (a) (x + 3) 5 and (b) (2p − q) 4 .
Answer:
Question 9.
Find the coefficient of x 5 in the expansion of (x − 3) 7 .
Answer:
Question 10.
Find the coefficient of x 3 in the expansion of (2x + 5) 8 .
Answer:
Permutations and Combinations 10.5 Exercises
Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
An arrangement of objects in which order is important is called a(n) __________.
Answer:
Question 2.
WHICH ONE DOESN’T BELONG?
Which expression does not belong with the other three? Explain your reasoning.
Answer:
Monitoring Progress and Modeling with Mathematics
In Exercises 3–8, find the number of ways you can arrange (a) all of the letters and (b) 2 of the letters in the given word.
Question 3.
AT
Answer:
Question 5.
ROCK
Answer:
Question 7.
FAMILY
Answer:
Question 8.
FLOWERS
Answer:
In Exercises 9–16, evaluate the expression.
Question 9.
_{5}P_{2}
Answer:
Question 11.
_{9}P_{1}
Answer:
Question 13.
_{8}P_{6}
Answer:
Question 15.
_{30}P_{2}
Answer:
Question 17.
PROBLEM SOLVING
Eleven students are competing in an art contest. In how many different ways can the students finish first, second, and third?
Answer:
Question 18.
PROBLEM SOLVING
Six friends go to a movie theater. In how many different ways can they sit together in a row of 6 empty seats?
Answer:
Question 19.
PROBLEM SOLVING
You and your friend are 2 of 8 servers working a shift in a restaurant. At the beginning of the shift, the manager randomly assigns one section to each server. Find the probability that you are assigned Section 1 and your friend is assigned Section 2.
Answer:
Question 20.
PROBLEM SOLVING
You make 6 posters to hold up at a basketball game. Each poster has a letter of the word TIGERS. You and 5 friends sit next to each other in a row. The posters are distributed at random. Find the probability that TIGERS is spelled correctly when you hold up the posters.
Answer:
In Exercises 21–24, count the possible combinations of r letters chosen from the given list.
Question 21.
A, B, C, D r = 3
Answer:
Question 22.
L, M, N, O r = 2
Answer:
Question 23.
U, V, W, X, Y, Z r = 3
Answer:
Question 24.
D, E, F, G, H r = 4
Answer:
In Exercises 25–32, evaluate the expression.
Question 25.
_{5}C_{1}
Answer:
Question 27.
_{9}C_{9}
Answer:
Question 29.
_{12}C_{3}
Answer:
Question 31.
_{15}C_{8}
Answer:
Question 33.
PROBLEM SOLVING
Each year, 64 golfers participate in a golf tournament. The golfers play in groups of 4.How many groups of 4 golfers are possible?
Answer:
Question 34.
PROBLEM SOLVING
You want to purchase vegetable dip for a party. A grocery store sells 7 different flavors of vegetable dip. You have enough money to purchase 2 flavors. How many combinations of 2 flavors of vegetable dip are possible?
Answer:
ERROR ANALYSIS In Exercises 35 and 36, describe and correct the error in evaluating the expression.
Question 35.
Answer:
Question 36.
Answer:
REASONING In Exercises 37–40, tell whether the question can be answered using permutations or combinations. Explain your reasoning. Then answer the question.
Question 37.
To complete an exam, you must answer 8 questions from a list of 10 questions. In how many ways can you complete the exam?
Answer:
Question 38.
Ten students are auditioning for 3 different roles in aplay. In how many ways can the 3 roles be filled?
Answer:
Question 39.
Fifty-two athletes are competing in a bicycle race. In how many orders can the bicyclists finish first, second, and third? (Assume there are no ties.)
Answer:
Question 40.
An employee at a pet store needs to catch 5 tetras in an aquarium containing 27 tetras. In how many groupings can the employee capture 5 tetras?
Answer:
Question 41.
CRITICAL THINKING
Compare the quantities _{50}C_{9} and _{50}C_{41} without performing any calculations. Explain your reasoning.
Answer:
Question 42.
CRITICAL THINKING
Show that each identity is true for any whole numbers r and n, where 0 ≤ r ≤ n.
a. _{n}C_{n} = 1
b. _{n}C_{r} = _{n}C_{n-r} − r
c. _{n+1}C_{r} = _{n}C_{r} + _{n}C_{r-1}
Answer:
Question 43.
REASONING
Consider a set of 4 objects.
a. Are there more permutations of all 4 of the objects or of 3 of the objects? Explain your reasoning.
b. Are there more combinations of all 4 of the objects or of 3 of the objects? Explain your reasoning.
c. Compare your answers to parts (a) and (b).
Answer:
Question 44.
OPEN-ENDED
Describe a real-life situation where the number of possibilities is given by _{5}P_{2}. Then describe a real-life situation that can be modeled by _{5}C_{2}.
Answer:
Question 45.
REASONING
Complete the table for each given value of r. Then write an inequality relating _{n}P_{r} and _{n}C_{r}. Explain your reasoning.
Answer:
Question 46.
REASONING
Write an equation that relates _{n}P_{r} and _{n}C_{r}. Then use your equation to find and interpret the value of (frac<182^
Answer:
Question 47.
PROBLEM SOLVING
You and your friend are in the studio audience on a television game show. From an audience of 300 people, 2 people are randomly selected as contestants. What is the probability that you and your friend are chosen?
Answer:
Question 48.
PROBLEM SOLVING
You work 5 evenings each week at a bookstore. Your supervisor assigns you 5 evenings at random from the 7 possibilities. What is the probability that your schedule does not include working on the weekend?
Answer:
REASONING In Exercises 49 and 50, find the probability of winning a lottery using the given rules. Assume that lottery numbers are selected at random.
Question 49.
You must correctly select 6 numbers, each an integer from 0 to 49. The order is not important.
Answer:
Question 50.
You must correctly select 4 numbers, each an integer from 0 to 9. The order is important.’
Answer:
In Exercises 51–58, use the Binomial Theorem to write the binomial expansion.
Question 51.
(x + 2) 3
Answer:
Question 52.
(c − 4) 5
Answer:
Question 53.
(a + 3b) 4
Answer:
Question 54.
(4p − q) 6
Answer:
Question 55.
(w 3 − 3) 4
Answer:
Question 56.
(2s 4 + 5) 5
Answer:
Question 57.
(3u + v 2 ) 6
Answer:
Question 58.
(x 3 − y 2 ) 4
Answer:
In Exercises 59–66, use the given value of n to find the coefficient of xn in the expansion of the binomial.
Question 59.
(x − 2) 10 , n = 5
Answer:
Question 60.
(x − 3) 7 , n = 4
Answer:
Question 61.
(x 2 − 3) 8 , n = 6
Answer:
Question 62.
(3x + 2) 5 , n = 3
Answer:
Question 63.
(2x + 5) 12 , n = 7
Answer:
Question 64.
(3x − 1) 9 , n = 2
Answer:
Question 65.
((frac<1><2>)x − 4 ) 11 , n = 4
Answer:
Question 66.
((frac<1><4>)x + 6 ) 6 , n = 3
Answer:
Question 67.
REASONING
Write the eighth row of Pascal’s Triangle as combinations and as numbers.
Answer:
Question 68.
PROBLEM SOLVING
The first four triangular numbers are 1, 3, 6, and 10.
a. Use Pascal’s Triangle to write the first four triangular numbers as combinations.
b. Use your result from part (a) to write an explicit rule for the nth triangular number T_{n}.
Answer:
Question 69.
MATHEMATICAL CONNECTIONS
A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. Consider a convex polygon with n sides.
a. Use the combinations formula to write an expression for the number of diagonals in an n-sided polygon.
b. Use your result from part (a) to write a formula for the number of diagonals of an n-sided convex polygon.
Answer:
Question 70.
PROBLEM SOLVING
You are ordering a burrito with 2 main ingredients and 3 toppings. The menu below shows the possible choices. How many different burritos are possible?
Answer:
Question 71.
PROBLEM SOLVING
You want to purchase 2 different types of contemporary music CDs and 1 classical music CD from the music collection shown. How many different sets of music types can you choose for your purchase?
Answer:
Question 72.
PROBLEM SOLVING
Every student in your history class is required to present a project in front of the class. Each day, 4 students make their presentations in an order chosen at random by the teacher. You make your presentation on the first day.
a. What is the probability that you are chosen to be the first or second presenter on the first day?
b. What is the probability that you are chosen to be the second or third presenter on the first day? Compare your answer with that in part (a).
Answer:
Question 73.
PROBLEM SOLVING
The organizer of a cast party for a drama club asks each of the 6 cast members to bring 1 food item from a list of 10 items. Assuming each member randomly chooses a food item to bring, what is the probability that at least 2 of the 6 cast members bring the same item?
Answer:
Question 74.
HOW DO YOU SEE IT?
A bag contains one green marble, one red marble, and one blue marble. The diagram shows the possible outcomes of randomly drawing three marbles from the bag without replacement.
a. How many combinations of three marbles can be drawn from the bag? Explain.
b. How many permutations of three marbles can be drawn from the bag? Explain.
Answer:
Question 75.
PROBLEM SOLVING
You are one of 10 students performing in a school talent show. The order of the performances is determined at random. The first 5 performers go on stage before the intermission.
a. What is the probability that you are the last performer before the intermission and your rival performs immediately before you?
b. What is the probability that you are not the first performer?
Answer:
Question 76.
THOUGHT PROVOKING
How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3, and 4? Repetition of digits is allowed.
Answer:
Question 77.
PROBLEM SOLVING
Consider a standard deck of 52 playing cards. The order in which the cards are dealt for a “hand” does not matter.
a. How many different 5-card hands are possible?
b. How many different 5-card hands have all 5 cards of a single suit?
Answer:
Question 78.
PROBLEM SOLVING
There are 30 students in your class. Your science teacher chooses 5 students at random to complete a group project. Find the probability that you and your 2 best friends in the science class are chosen to work in the group. Explain how you found your answer.
Answer:
Question 79.
PROBLEM SOLVING
Follow the steps below to explore a famous probability problem called the birthday problem. (Assume there are 365 equally likely birthdays possible.)
a. What is the probability that at least 2 people share the same birthday in a group of 6 randomly chosen people? in a group of 10 randomly chosen people?
b. Generalize the results from part (a) by writing a formula for the probability P(n) that at least 2 people in a group of n people share the same birthday. (Hint: Use _{n}P_{r} notation in your formula.)
c. Enter the formula from part (b) into a graphing calculator. Use the table feature to make a table of values. For what group size does the probability that at least 2 people share the same birthday first exceed 50%?
Answer:
Maintaining Mathematical Proficiency
Question 80.
A bag contains 12 white marbles and 3 black marbles. You pick 1 marble at random. What is the probability that you pick a black marble?
Answer:
Question 81.
The table shows the result of flipping two coins 12 times. For what outcome is the experimental probability the same as the theoretical probability?
Answer:
Lesson 10.6 Binomial Distributions
Essential Question How can you determine the frequency of each outcome of an event?
EXPLORATION 1
Analyzing Histograms
Work with a partner. The histograms show the results when n coins are flipped.
a. In how many ways can 3 heads occur when 5 coins are flipped?
b. Draw a histogram that shows the numbers of heads that can occur when 6 coins are flipped.
c. In how many ways can 3 heads occur when 6 coins are flipped?
EXPLORATION 2
Determining the Number of Occurrences
Work with a partner.
a. Complete the table showing the numbers of ways in which 2 heads can occur when n coins are flipped.
b. Determine the pattern shown in the table. Use your result to find the number of ways in which 2 heads can occur when 8 coins are flipped.
Communicate Your Answer
Question 3.
How can you determine the frequency of each outcome of an event?
Answer:
Question 4.
How can you use a histogram to find the probability of an event?
Answer:
Monitoring Progress
An octahedral die has eight sides numbered 1 through 8. Let x be a random variable that represents the sum when two such dice are rolled.
Question 1.
Make a table and draw a histogram showing the probability distribution for x.
Answer:
Question 2.
What is the most likely sum when rolling the two dice?
Answer:
Question 3.
What is the probability that the sum of the two dice is at most 3?
Answer:
According to a survey, about 85% of people ages 18 and older in the U.S. use the Internet or e-mail. You ask 4 randomly chosen people (ages 18 and older) whether they use the Internet or e-mail.
Question 4.
Draw a histogram of the binomial distribution for your survey.
Answer:
Question 5.
What is the most likely outcome of your survey?
Answer:
Question 6.
What is the probability that at most 2 people you survey use the Internet or e-mail?
Answer:
Binomial Distributions 10.6 Exercises
Vocabulary and Core Concept Check
Question 1.
VOCABULARY
What is a random variable?
Answer:
Question 2.
WRITING
Give an example of a binomial experiment and describe how it meets the conditions of a binomial experiment.
Answer:
Monitoring Progress and Modeling with Mathematics
In Exercises 3–6, make a table and draw a histogram showing the probability distribution for the random variable.
Question 3.
x = the number on a table tennis ball randomly chosen from a bag that contains 5 balls labeled “1,” 3 balls labeled “2,” and 2 balls labeled “3.”
Answer:
Question 4.
c = 1 when a randomly chosen card out of a standard deck of 52 playing cards is a heart and c = 2 otherwise.
Answer:
Question 5.
w = 1 when a randomly chosen letter from the English alphabet is a vowel and w = 2 otherwise.
Answer:
Question 6.
n = the number of digits in a random integer from 0 through 999.
Answer:
In Exercises 7 and 8, use the probability distribution to determine (a) the number that is most likely to be spun on a spinner, and (b) the probability of spinning an even number.
Question 7.
Answer:
Question 8.
Answer:
USING EQUATIONS In Exercises 9–12, calculate the probability of flipping a coin 20 times and getting the given number of heads.
Question 9.
1
Answer:
Question 11.
18
Answer:
Question 13.
MODELING WITH MATHEMATICS
According to a survey, 27% of high school students in the United States buy a class ring. You ask 6 randomly chosen high school students whether they own a class ring.
a. Draw a histogram of the binomial distribution for your survey.
b. What is the most likely outcome of your survey?
c. What is the probability that at most 2 people have a class ring?
Answer:
Question 14.
MODELING WITH MATHEMATICS
According to a survey, 48% of adults in the United States believe that Unidentified Flying Objects (UFOs) are observing our planet. You ask 8 randomly chosen adults whether they believe UFOs are watching Earth.
a. Draw a histogram of the binomial distribution for your survey.
b. What is the most likely outcome of your survey?
c. What is the probability that at most 3 people believe UFOs are watching Earth?
Answer:
ERROR ANALYSIS In Exercises 15 and 16, describe and correct the error in calculating the probability of rolling a 1 exactly 3 times in 5 rolls of a six-sided die.
Question 15.
Answer:
Question 16.
Answer:
Question 17.
MATHEMATICAL CONNECTIONS
At most 7 gopher holes appear each week on the farm shown. Let x represent how many of the gopher holes appear in the carrot patch. Assume that a gopher hole has an equal chance of appearing at any point on the farm.
a. Find P(x) for x= 0, 1, 2, . . . , 7.
b. Make a table showing the probability distribution for x.
c. Make a histogram showing the probability distribution for x.
Answer:
Question 18.
HOW DO YOU SEE IT?
Complete the probability distribution for the random variable x. What is the probability the value of x is greater than 2?
Answer:
Question 19.
MAKING AN ARGUMENT
The binomial distribution shows the results of a binomial experiment. Your friend claims that the probability p of a success must be greater than the probability 1 −p of a failure. Is your friend correct? Explain your reasoning.
Answer:
Question 20.
THOUGHT PROVOKING
There are 100 coins in a bag. Only one of them has a date of 2010. You choose a coin at random, check the date, and then put the coin back in the bag. You repeat this 100 times. Are you certain of choosing the 2010 coin at least once? Explain your reasoning.
Answer:
Question 21.
MODELING WITH MATHEMATICS
Assume that having a male and having a female child are independent events, and that the probability of each is 0.5.
a. A couple has 4 male children. Evaluate the validity of this statement: “The first 4 kids were all boys, so the next one will probably be a girl.”
b. What is the probability of having 4 male children and then a female child?
c. Let x be a random variable that represents the number of children a couple already has when they have their first female child. Draw a histogram of the distribution of P(x) for 0 ≤ x ≤ 10. Describe the shape of the histogram.
Answer:
Question 22.
CRITICAL THINKING
An entertainment system has n speakers. Each speaker will function properly with probability p, independent of whether the other speakers are functioning. The system will operate effectively when at least 50% of its speakers are functioning. For what values of p is a 5-speaker system more likely to operate than a 3-speaker system?
Answer:
Maintaining Mathematical Proficiency
List the possible outcomes for the situation.
Question 23.
guessing the gender of three children
Answer:
Question 24.
picking one of two doors and one of three curtains
Answer:
Probability Performance Task: A New Dartboard
10.4–10.6 What Did You Learn?
Core Vocabulary
Core Concepts
Section 10.4
Probability of Compound Events, p. 564
Section 10.5Permutations, p. 571Combinations, p. 572
The Binomial Theorem, p. 574
Section 10.6
Probability Distributions, p. 580
Binomial Experiments, p. 581
Mathematical Practices
Question 1.
How can you use diagrams to understand the situation in Exercise 22 on page 568?
Answer:
Question 2.
Describe a relationship between the results in part (a) and part (b) in Exercise 74 on page 578.
Answer:
Question 3.
Explain how you were able to break the situation into cases to evaluate the validity of the statement in part (a) of Exercise 21 on page 584.
Answer:
Performance Task A New Dartboard
You are a graphic artist working for a company on a new design for the board in the game of darts. You are eager to begin the project, but the team cannot decide on the terms of the game. Everyone agrees that the board should have four colors. But some want the probabilities of hitting each color to be equal, while others want them to be different. You offer to design two boards, one for each group. How do you get started? How creative can you be with your designs?
To explore the answers to these questions and more, go to BigIdeasMath.com.
Probability Chapter Review
10.1 Sample Spaces and Probability (pp. 537–544)
Question 1.
A bag contains 9 tiles, one for each letter in the word HAPPINESS. You choose a tile at random. What is the probability that you choose a tile with the letter S? What is the probability that you choose a tile with a letter other than P?
Answer:
Question 2.
You throw a dart at the board shown. Your dart is equally likely to hit any point inside the square board. Are you most likely to get 5 points, 10 points, or 20 points?
Answer:
10.2 Independent and Dependent Events (pp. 545–552)
Find the probability of randomly selecting the given marbles from a bag of 5 red, 8 green, and 3 blue marbles when (a) you replace the first marble before drawing the second, and (b) you do not replace the first marble. Compare the probabilities.
Question 3.
red, then green
Answer:
Question 4.
blue, then red
Answer:
Question 5.
green, then green
Answer:
10.3 Two-Way Tables and Probability (pp. 553–560)
Question 6.
What is the probability that a randomly selected resident who does not support the project in the example above is from the west side?
Answer:
Question 7.
After a conference, 220 men and 270 women respond to a survey. Of those, 200 men and 230 women say the conference was impactful. Organize these results in a two-way table. Then find and interpret the marginal frequencies.
Answer:
10.4 Probability of Disjoint and Overlapping Events (pp. 563–568)
Question 8.
Let A and B be events such that P(A) = 0.32, P(B) = 0.48, and P(A and B) = 0.12. Find P(A or B).
Answer:
Question 9.
Out of 100 employees at a company, 92 employees either work part time or work 5 days each week. There are 14 employees who work part time and 80 employees who work 5 days each week. What is the probability that a randomly selected employee works both part time and 5 days each week?
Answer:
10.5 Permutations and Combinations (pp. 569–578)
Evaluate the expression.
Question 10.
_{7}P_{6}
Answer:
Question 14.
Use the Binomial Theorem to write the expansion of (2x + y 2 ) 4 .
Answer:
Question 15.
A random drawing will determine which 3 people in a group of 9 will win concert tickets. What is the probability that you and your 2 friends will win the tickets?
Answer:
10.6 Binomial Distributions (pp. 579–584)
Question 16.
Find the probability of flipping a coin 12 times and getting exactly 4 heads.
Answer:
Question 17.
A basketball player makes a free throw 82.6% of the time. The player attempts 5 free throws. Draw a histogram of the binomial distribution of the number of successful free throws. What is the most likely outcome?
Answer:
Probability Chapter Test
You roll a six-sided die. Find the probability of the event described. Explain your reasoning.
Question 1.
You roll a number less than 5.
Answer:
Question 2.
You roll a multiple of 3.
Answer:
Evaluate the expression.
Question 3.
_{7}P_{2}
Answer:
Question 7.
Use the Binomial Theorem to write the binomial expansion of (x + y 2 ) 5 .
Answer:
Question 8.
You find the probability P(A or B) by using the equation P(A or B) = P(A) + P(B) − P(A and B). Describe why it is necessary to subtract P(A and B) when the events A and B are overlapping. Then describe why it is not necessary to subtract P(A and B) when the events A and B are disjoint.
Answer:
Question 9.
Is it possible to use the formula P(A and B) =P(A) • P(B|A) when events A and B are independent? Explain your reasoning.
Answer:
Question 10.
According to a survey, about 58% of families sit down for a family dinner at least four times per week. You ask 5 randomly chosen families whether they have a family dinner at least four times per week.
a. Draw a histogram of the binomial distribution for the survey.
b. What is the most likely outcome of the survey?
c. What is the probability that at least 3 families have a family dinner four times per week?
Answer:
Question 11.
You are choosing a cell phone company to sign with for the next 2 years. The three plans you consider are equally priced. You ask several of your neighbors whether they are satisfied with their current cell phone company. The table shows the results. According to this survey, which company should you choose?
Answer:
Question 12.
The surface area of Earth is about 196.9 million square miles. The land area is about 57.5 million square miles and the rest is water. What is the probability that a meteorite that reaches the surface of Earth will hit land? What is the probability that it will hit water?
Answer:
Question 13.
Consider a bag that contains all the chess pieces in a set, as shown in the diagram.
a. You choose one piece at random. Find the probability that you choose a black piece or a queen.
b. You choose one piece at random, do not replace it, then choose a second piece at random. Find the probability that you choose a king, then a pawn.
Answer:
Question 14.
Three volunteers are chosen at random from a group of 12 to help at a summer camp.
a. What is the probability that you, your brother, and your friend are chosen?
b. The first person chosen will be a counselor, the second will be a lifeguard, and the third will be a cook. What is the probability that you are the cook, your brother is the lifeguard, and your friend is the counselor?
Answer:
Probability Cumulative Assessment
Question 1.
According to a survey, 63% of Americans consider themselves sports fans. You randomly select 14 Americans to survey.
a. Draw a histogram of the binomial distribution of your survey.
b. What is the most likely number of Americans who consider themselves sports fans?
c. What is the probability at least 7 Americans consider themselves sports fans?
Answer:
Question 2.
Order the acute angles from smallest to largest. Explain your reasoning.
Answer:
Question 3.
You order a fruit smoothie made with 2 liquid ingredients and 3 fruit ingredients from the menu shown. How many different fruit smoothies can you order?
Answer:
Question 4.
Which statements describe the transformation of the graph of f(x) = x 3 − x represented by g(x) = 4(x − 2) 3 − 4(x − 2)?
A. a vertical stretch by a factor of 4
B. a vertical shrink by a factor of (frac<1><4>)
C. a horizontal shrink by a factor of (frac<1><4>)
D. a horizontal stretch by a factor of 4
E. a horizontal translation 2 units to the right
F. a horizontal translation 2 units to the left
Answer:
Question 5.
Use the diagram to explain why the equation is true. P(A) + P(B) = P(A or B) + P(A and B)
Answer:
Question 6.
For the sequence (-frac<1><2>,-frac<1><4>,-frac<1><6>,-frac<1><8>, ldots) describe the pattern, write the next term, graph the first five terms, and write a rule for the nth term.
Answer:
Question 7.
A survey asked male and female students about whether they prefer to take gym class or choir. The table shows the results of the survey.
a. Complete the two-way table.
b. What is the probability that a randomly selected student is female and prefers choir?
c. What is the probability that a randomly selected male student prefers gym class?
Answer:
Question 8.
The owner of a lawn-mowing business has three mowers. As long as one of the mowers is working, the owner can stay productive. One of the mowers is unusable 10% of the time, one is unusable 8% of the time, and one is unusable 18% of the time.
a. Find the probability that all three mowers are unusable on a given day.
b. Find the probability that at least one of the mowers is unusable on a given day.
c. Suppose the least-reliable mower stops working completely. How does this affect the probability that the lawn-mowing business can be productive on a given day?
Answer:
Question 9.
Write a system of quadratic inequalities whose solution is represented in the graph.
Answer:
6.1: Sample Spaces and Probability - Mathematics
The course will introduce the basic notion of probability theory and its application to statistics. The focus will be on the discussion of applications.
The text that will be used is:
Jay L. Devore, Probability and Statistics, 8th or 9th ed., Thomson
The syllabus can be found here. You can see how the class will develop on the webpage of the previous time I teached it.
There will be two midterm.
The exercise listed are for HW collection. I will collect them every two weeks and grade 2 or 3 exercises among the one assigned. In case of differences between the 9th and 8th editions of the book I will indicate in square brackets the number relative to the 8th edition.
The final grade will be based on the following rules: 40% final, 40% midterms,20% HW. Curving will be done on the final result.
The first midterm is tentatively scheduled for Wednesday February 21 and the second on Wednesday March 28.
Arguments covered.
- Axioms, Interpretations and Properties of Probabilities
- Probability Distributions for Discrete Random Variables
- Example of Discrete Random Variables
- Continuous Random Variables and Probability Density Functions
- Example of Continuous Random Variables
- The central limit theorem
- Jointly Distributed Random Variables
- Population, Sample and Processes
- Point Estimation
- Statistical Intervals
- Test of Hypotheses
- Simple Linear Regression (time permitting)
- 1.1 (Population, Sample and Processes)
- 1.2 (Pictorial and Tabular methods in Descriptive Statistics)
- 1.3 (Measure of Location)
- 1.4 (Measure of Variability)
- 2.1 (Sample Spaces and Events)
- 2.2 (Axioms, Interpretations and Properties of Probabilities)
First HW due on January 29.
- 3.1 (Random Variables)
- 3.2 (Probability Distributions for Discrete Random Variables)
- 3.3 (Expected Values of Discrete Random Variable)
- 3.4 (The Binomial Probability Distribution)
- 3.5 (Hypergeometric Distribution)
- 3.6 (The Poisson Probability Distribution)
- 4.1 (Continuous Random Variables and Probability Density Functions)
- 4.2 (Cumulative Distribution Functions and Expected Values)
- 4.3 (The Normal Distribution)
- 4.4 (The Exponential Distribution)
The first midterm will be on February 28. The midterm will cover the material up to section 4.4.
Preparation material for the first midterm:
- 4.6 (Probability Plots)
- 5.1 (Jointly Distributed Random Variables)
- 5.2 (Expected Values,Covariance and Correlation)
- 5.5 (The Distribution of a Linear Combination)
- 5.3 (Statistics and their distribution)
- 5.4 (The Distribution of the Sample Mean)
- (5.3) 37, 41, 42
- (5.4) 48, 49, 53, 56
- (5.5) 59, 64, 68
The second midterm will be on Wednesday April 4. It will cover all the material up to Chapter 5 included. You may use a scientific calculator but no laptop or calculator able to do symbolic differentiation or integration. No cheat sheet will be allowed.
Preparation material for the second midterm:
Here is the text of the Optional Makeup for the second Midterm. Check it out and let me know if you have question. It is due Monday April 23 before class. You can submit it at the beginning of Monday class, via email or by sliding it under my office door. Your submission must contain the first page signed
The exercise listed below are intended to help your preparation toward the final. They will not be collected.
Big Ideas Math Book Geometry Answer Key Chapter 12 Probability
Make the most out of the Big Ideas Math Geometry Answer Key for Cha 12 Probability and score better grades in the exams. Simply tap on the quick links available and prepare the respective topics available as and when you need them. Big Ideas Math Geometry Answers Chapter 12 Probability available here covers questions from the Topics Sample Spaces, Independent and Dependent Events, Disjoint and Overlapping Events, Binomial Distributions, Probability Test, etc.
Lesson: 1 Sample Spaces and Probability
Lesson: 2 Independent and Dependent Events
Lesson: 3 Two-Way Tables and Probability
Lesson: 4 Probability of Disjoint and Overlapping Events
Lesson: 5 Permutations and Combinations
Lesson: 6 Binomial Distributions
Chapter: 12 – Probability
Probability Maintaining Mathematical Proficiency
Write and solve a proportion to answer the question.
Question 1.
What percent of 30 is 6?
Answer: 20
Explanation:
100% = 30
x% = 6
100% = 30(1)
x% = 6(2)
100%/x% = 30/6
Taking the inverse of both sides
x%/100% = 6/30
x = 20%
Thus 6 is 20% of 30.
Question 2.
What number is 68% of 25?
Answer: 17
Explanation:
68% × 25
(68 ÷ 100) × 25
(68 × 25) ÷ 100
1700 ÷ 100 = 17
Question 3.
34.4 is what percent of 86?
Answer: 40
Explanation:
100% = 86
x% = 34.4
100% = 86(1)
x% = 34.4(2)
100%/x% = 86/34.4
Taking the inverse of both sides
x%/100% = 34.4/86
x = 40%
Therefore, 34.4 is 40% of 86.
Display the data in a histogram.
Question 4.
Answer:
Question 5.
ABSTRACT REASONING
You want to purchase either a sofa or an arm chair at a furniture store. Each item has the same retail price. The sofa is 20% off. The arm chair is 10% off. and you have a coupon to get an additional 10% off the discounted price of the chair. Are the items equally priced after the discounts arc applied? Explain.
Answer: Yes
Explanation:
You want to purchase either a sofa or an armchair at a furniture store.
Each item has the same retail price. The sofa is 20% off. The armchair is 10% off. and you have a coupon to get an additional 10% off the discounted price of the chair.
The price of armchair and sofa are the same.
If you add 10% to chair the discount for the chair and sofa will be the same.
10% + 10% = 20%
Probability Monitoring Progress
In Exercises 1 and 2, describe the event as unlikely, equally likely to happen or not happen, or likely. Explain your reasoning.
Question 1.
The oldest child in a family is a girl.
Answer:
Question 2.
The two oldest children in a family with three children are girls.
Answer:
Question 3.
Give an example of an event that is certain to occur.
Answer:
If A and B are independent event
P(A) = 1/2
P(B) = 1/5
P(A and B) = P(A) × P(B)
= 1/2 × 1/5
= 1/10
12.1 Sample Spaces and Probability
Exploration 1
Finding the Sample Space of an Experiment
Work with a partner: In an experiment, three coins are flipped. List the possible outcomes in the sample space of the experiment.
Answer:
The number of different outcomes when three coins are tossed is 2 × 2 × 2 = 8.
All 8 possible outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH and TTT.
Exploration 2
Finding the Sample Space of an Experiment
Work with a partner: List the possible outcomes in the sample space of the experiment.
a. One six-sided die is rolled.
Answer: 6 possible outcomes
b. Two six-sided die is rolled.
Answer:
Rolling two six-sided dice: Each die has 6 equally likely outcomes, so the sample space is 6 . 6 or 36 equally likely outcomes.
Exploration 3
Finding the Sample Space of an Experiment
Work with a partner: In an experiment, a spinner is spun.
a. How many ways can you spin a 1? 2? 3? 4? 5?
Answer: 1, 2, 3, 2, 4
b. List the sample space.
Answer: 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 5
c. What is the total number of outcomes?
Answer: 12
Finding the Sample Space of an Experiment
Work with a partner: In an experiment, a bag contains 2 blue marbles and 5 red marbles. Two marbles arc drawn from the bag.
a. How many ways can you choose two blue? a red then blue? a blue then red? two red?
Answer: BB – 2, RB – 10, BR – 10, RR – 20
b. List the sample space.
Answer:
c. What is the total number of outcomes?
Answer: 42
Question 5.
How can you list the possible outcomes in the sample space of an experiment?
Answer:
There are four possible outcomes for each spin: red, blue, yellow, green. Then, multiply the number of outcomes by the number of spins. June flipped the coin three times. The answer is there are 12 outcomes in the sample space.
Question 6.
For Exploration 3, find the ratio of the number of each possible outcome to the total number of outcomes. Then find the sum of these ratios. Repeat for Exploration 4. What do you observe?
LOOKING FOR A PATTERN
To be proficient in math, you need to look closely to discern a pattern or structure.
Answer:
Lesson 12.1 Sample Spaces and Probability
Monitoring Progress
Find the number of possible outcomes in the sample space. Then list the possible outcomes.
Question 1.
You flip two coins.
Answer: 4
Explanation:
When we flip two coins simultaneoulsy then the possible outcomes will be (H, H), (T, T), (T, H), (H, T)
where H represents heads
T represents tails.
Thus the possible outcomes are 2² = 4
Question 2.
You flip two Coins and roll a six-sided die.
Answer: 6 × 2² = 24
Explanation:
We roll a die and flip two coins. We have to find the number of possible outcomes in this space. Also we have to list the possible outcomes.
1 =
2 =
3 =
4 =
5 =
6 =
On the other hand, using H for Heads and T for Tails we can list the outcomes.
(1, H, H), (2, H, H), (3, H, H), (4, H, H), (5, H, H), (6, H, H)
(1, T, H), (2, T, H), (3, T, H), (4, T, H), (5, T, H), (6, T, H)
(1, H, T), (2, H, T), (3, H, T), (4, H, T), (5, H, T), (6, H, T)
(1, T, T), (2, T, T), (3, T, T), (4, T, T), (5, T, T), (6, T, T)
Therefore, we can conclude that the number of all possible outcomes is
6 × 2² = 24
Question 3.
You flip a coin and roll a six-sided die. What is the probability that the coin shows tails and the die shows 4?
Answer:
The sample space has 12 possible outcomes.
Heads, 1
Heads, 2
Heads, 3
Heads, 4
Heads, 5
Heads, 6
Tails, 1
Tails, 2
Tails, 3
Tails, 4
Tails, 5
Tails, 6
Probability that the coin shows tails and the die shows 4 is 4/12 = 1/3
Question 8.
In Example 4, are you more likely to get 10 points or 5 points?
Answer:
10: 0.09
5: (5 – 10)/324
= (36π – 9π)/324
= 27π/324
= 0.26
Question 9.
In Example 4, are you more likely to score points (10, 5, or 2) or get 0 points?
Answer:
2: (2 – 5)/324
= (81π – 36π)/324
= 45π/324
= 0.43
0.09 + 0.26 + 0.43 = 0.78
More likely to get 2 points.
Question 10.
In Example 5, for which color is the experimental probability of stopping on the color greater than the theoretical probability?
Answer:
9/20 = 0.45
Question 11.
In Example 6, what is the probability that a pet-owning adult chosen at random owns a fish?
Answer:
146/1328 = 73/664 = 0.11
Exercise 12.1 Sample Spaces and Probability
Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
A number that describes the likelihood of an event is the ___________ of the event.
Answer:
A number that describes the likelihood of an event is the Probability of the event.
Question 2.
WRITING
Describe the difference between theoretical probability and experimental probability.
Answer: Experimental probability is the result of an experiment. Theoretical probability is what is expected to happen.
Monitoring Progress and Modeling with Mathematics
In Exercises 3 – 6. find the number of possible outcomes in the sample space. Then list the possible outcomes.
Question 3.
You roll a die and flip three coins.
Answer:
Question 4.
You flip a coin and draw a marble at random from a hag containing two purple marbles and one while marble.
Answer:
Given data,
You flip a coin and draw a marble at random from a hag containing two purple marbles and one while marble.
the probability of getting a purple marble = 2/3
the probability of getting a white marble = 1/3
Question 5.
A bag contains four red cards numbered 1 through 4, four white cards numbered 1 through 4, and four black cards numbered 1 through 4. You choose a card at random.
Answer:
Question 6.
You draw two marbles without replacement from a bag containing three green marbles and four black marbles.
Answer:
In all there are 7 marbles when you first grab a marble, after that you take one marble away then you have 6 marbles to choose.
7 × 6 = 42
42 possible outcomes: GG, GG, GB, GB, GB, GB, GG, GG, GB, GB, GB, GB, GG, GG, GB, GB, GB, GB, BG, BG, BG, BB, BB, BB, BG, BG, BG, BB, BB, BB, BG, BG, BG, BB, BB, BB, BG, BG, BG, BB, BB, BB, BB.
Question 7.
PROBLEM SOLVING
A game show airs on television five days per week. Each day, a prize is randomly placed behind one of two doors. The contestant wins the prize by selecting the correct door. What is the probability that exactly two of the five contestants win a prize during a week?
Answer:
Question 8.
PROBLEM SOLVING
Your friend has two standard decks of 52 playing cards and asks you to randomly draw one card from each deck. What is the probability that you will draw two spades?
Answer:
You have two decks of 52 cards and in a normal deck, there are 13 cards of each suit.
So there are 13 spades in the deck.
Therefore the probability of you drawing a spade is 13 out of all the 52 cards or (frac<13><52>), which can be reduced to (frac<1><4>). You do this two times with different decks that are exactly the same so you multiply (frac<1><4>) times (frac<1><4>). 1 times 1 is 1 and 4 times 4 is 16, so it is (frac<1><16>).
You can turn this into a percentage by dividing 1 by 16 and moving the decimal place to places to the right.
6.25% is the probability that you will draw two spades.
Question 9.
PROBLEM SOLVING
When two six-sided dice are rolled, there are 36 possible outcomes. Find the probability that
(a) the sum is not 4 and
(b) the sum is greater than 5.
Answer:
Question 10.
PROBLEM SOLVING
The age distribution of a population is shown. Find the probability of each event.
a. A person chosen at random is at least 15 years old.
Answer: 80%
b. A person chosen at random is from 25 to 44 years old.
Answer: 13% + 13% = 26%
Question 11.
ERROR ANALYSIS
A student randomly, guesses the answers to two true-false questions. Describe and correct the error in finding the probability of the student guessing both answers correctly.
Answer:
Question 12.
ERROR ANALYSIS
A student randomly draws a number between 1 and 30. Describe and correct the error in finding the probability that the number drawn is greater than 4.
Answer:
The error is that the probability of the complement of the event is 4/30, not 3/30, because if you are looking for a sum greater than 4, than you subtract 1 by numbers less than or equal to 4 by the total amount of numbers, which is 30.
P(Sum is greater than 4)=1-P(Sum is less than or equal to 4)
1 – 2/15
= 13/15
Question 13.
MATHEMATICAL CONNECTIONS
You throw a dart at the board shown. Your dart is equally likely to hit any point inside the square board. What is the probability your dart lands in the yellow region?
Answer:
Question 14.
MATHEMATICAL CONNECTIONS
The map shows the length (in miles) of shoreline along the Gulf of Mexico for each state that borders the body of water. What is the probability that a ship coming ashore at a random point in the Gulf of Mexico lands in the given state?
a. Texas
Answer:
By using the above map we can solve the problem
Total length of the shoreline: 367 + 397 + 44 + 53 + 770 = 1631 miles
The probability of the ship landing in Texas: 367/1631 = 0.23
b. Alabama
Answer:
The probability of the ship landing in Alabama: 53/1631 = 0.03
c. Florida
Answer:
The probability of the ship landing in Florida: 770/1631 = 0.47
d. Louisiana
Answer:
The probability of the ship landing in Louisiana: 397/1631 = 0.24
Question 15.
DRAWING CONCLUSIONS
You roll a six-sided die 60 times. The table shows the results. For which number is the experimental probability of rolling the number the same as the theoretical probability?
Answer:
Question 16.
DRAWING CONCLUSIONS
A bag contains 5 marbles that are each a different color. A marble is drawn, its color is recorded, and then the marble is placed back in the hag. This process is repeated until 30 marbles have been drawn. The table shows the results. For which marble is the experimental probability of drawing the marble the same as the theoretical probability?
Answer:
Total number of marbles = 30
6/30 = 1/5
For black marble the experimental probability of drawing the marble the same as the theoretical probability.
Question 17.
REASONING
Refer to the spinner shown. The spinner is divided into sections with the same area.
a. What is the theoretical probability that the spinner stops on a multiple of 3?
b. You spin the spinner 30 times. If stops on a multiple of 3 twenty times. What is the experimental probability of Stopping on a multiple of 3?
c. Explain why the probability you found in part (b) is different than the probability you found in part (a).
Answer:
Question 18.
OPEN-ENDED
Describe a real-life event that has a probability of 0. Then describe a real-life event that has a probability of 1.
Answer:
The probability of rolling a 7 with a standard 6-sided die = 0
The probability of rolling a natural number with a standard 6-sided die = 1
Question 19.
DRAWING CONCLUSIONS
A survey of 2237 adults ages 18 and over asked which Sport 15 their favorite. The results are shown in the figure. What is the probability that an adult chosen at random prefers auto racing?
Answer:
Question 20.
DRAWING CONCLUSIONS
A survey of 2392 adults ages 18 and over asked what type of food they Would be most likely to choose at a restaurant. The results are shown in the figure. What is the probability that an adult chosen at random prefers Italian food?
Answer:
P(Italian) = 526/1196
= 263/1196
P(Italian) ≈ 22%
Question 21.
ANALYZING RELATIONSHIPS
Refer to the board in Exercise 13. Order the likelihoods that the dart lands in the given region from least likely to most likely.
A. green
B. not blue
C. red
D. not yellow
Answer:
Question 22.
ANALYZING RELATIONSHIPS
Refer to the chart below. Order the following events from least likely to most likely.
A. It rains on Sunday.
Answer:
80%
= 80/20 = 4/1
The ratio is 4:1
B. It does not rain on Saturday.
Answer:
100 – 30 = 70%
= 30/70 = 3 : 7
C. It rains on Monday.
Answer: 90%
= 90/10 = 9/1
= 9 : 1
D. It does not rain on Friday.
Answer:
100 – 5 = 95%
= 5/95 = 1/19
= 1 : 19
Question 23.
USING TOOLS
Use the figure in Example 3 to answer each question.
a. List the possible sums that result from rolling two six-sided dice.
b. Find the theoretical probability of rolling each sum.
c. The table below shows a simulation of rolling two six-sided dice three times. Use a random number generator to simulate rolling two six-sided dice 50 times. Compare the experimental probabilities of rolling each sum with the theoretical probabilities.
Answer:
Question 24.
MAKING AN ARGUMENT
You flip a coin three times. It lands on heads twice and on tails once. Your friend concludes that the theoretical probability of the coin landing heads up is P(heads up) = (frac<2><3>). Is your friend correct? Explain your reasoning.
Answer:
The friend is incorrect because the probability of heads here is (frac<2><3>), is the experimental probabiility of heads for this particular case, while its theoretical probability will always be (frac<1><2>).
Question 25.
MATHEMATICAL CONNECTIONS
A sphere fits inside a cube so that it touches each side, as shown. What is the probability a point chosen at random inside the cube is also inside the sphere ?
Answer:
Question 26.
HOW DO YOU SEE IT?
Consider the graph of f shown. What is the probability that the graph of y = f(x) + c intersects the x-axis when c is a randomly chosen integer from 1 to 6? Explain.
Answer:
Question 27.
DRAWING CONCLUSIONS
A manufacturer tests 1200 computers and finds that 9 of them have defects. Find the probability that a computer chosen at random has a defect. Predict the number of computers with defects in a shipment of 15,000 computers. Explain your reasoning.
Answer:
Question 28.
THOUGHT PROVOKING
The tree diagram shows a sample space. Write a probability problem that can be represented by the sample space. Then write the answer(s) to the problem.
Answer:
Maintaining Mathematical Proficiency
Simplify the expression. Write your answer using only positive exponents.
Question 29.
(frac<2 x^<3>>
Answer:
Question 31.
(frac <4 x^<9>y> <3 x^<3>y>)
Answer:
Question 33.
(3Pq) 4
Answer:
12.2 Independent and Dependent Events
Exploration 1
Identifying Independent and Dependent Events
Work with a partner: Determine whether the events are independent or dependent. Explain your reasoning.
REASONING ABSTRACTLY
To be proficient in math, you need to make sense of quantities and their relationships in problem situations.
a. Two six-sided dice are rolled.
Answer:
P = Number of outcomes that satisfy the requirements/Total number of possible outcomes
Probability for rolling two dice with the six sided dots such as 1, 2, 3, 4, 5 and 6 dots in each die.
When two dice are thrown simultaneously, thus the number of event can be 62 = 36 because each die has 1 to 6 number on its faces.
This is independent event
b. Six pieces of paper, numbered 1 through 6, are in a bag, Two pieces of paper are selected one at a time without replacement.
Answer:
P = Number of outcomes that satisfy the requirements/Total number of possible outcomes
When we pick up any paper the probability result is equal of 1/6
This is dependent event
Exploration 2
a. In Exploration 1(a), experimentally estimate the probability that the sum of the two numbers rolled is 7. Describe your experiment.
Answer:
(b). In Exploration 1 (b), experimentally estimate the probability that the sum of the two numbers selected is 7. Describe your experiment.
Answer:
Exploration 3
Finding Theoretical Probabilities
Work with a partner:
a. In Exploration 1(a), find the theoretical probability that the sum of the two numbers rolled is 7. Then compare your answer with the experimental probability you found in Exploration 2(a).
Answer:
For each of the possible outcomes add the numbers on the two dice and count how many times this sum is 7. If you do so you will find that the sum is 7 for 6 of the possible outcomes. Thus the sum is a 7 in 6 of the 36 outcomes and hence the probability of rolling a 7 is 6/36 = 1/6.
b. In Exploration 1(b). find the theoretical probability that the sum of the two numbers selected is 7. Then compare your answer with the experimental probability you found in Exploration 2(b).
Answer:
C. Compare the probabilities you obtained in parts (a) and (b).
Answer:
Question 4.
How can you determine whether two events are independent or dependent?
Answer:
Two events A and B are said to be independent if the fact that one event has occurred does not affect the probability that the other event will occur.
If whether or not one event occurs does affect the probability that the other event will occur, then the two events are said to be dependent.
Question 5.
Determine whether the events are independent or dependent. Explain your reasoning.
a. You roil a 4 on a six-sided die and spin red on a spinner.
Answer: Independent
b. Your teacher chooses a student to lead a group. chooses another student to lead a second group. and chooses a third student to lead a third group.
Answer: Dependent
Lesson 12.2 Independent and Dependent Events
Monitoring progress
Question 1.
In Example 1, determine whether guessing Question 1 incorrectly and guessing Question 2 correctly are independent events.
Answer:
Question 2.
In Example 2, determine whether randomly selecting a girl first and randomly selecting a boy second are independent events.
Answer:
Question 3.
In Example 3, what is the probability that you spin an even number and then an odd number?
Answer:
Question 4.
In Example 4, what is the probability that both hills are $1 hills?
Answer:
Question 5.
In Example 5, what is the probability that none of the cards drawn are hearts when (a) you replace each card, and (b) you do not replace each card? Compare the probabilities.
Answer:
(a) you replace each card,
P(A and B and C) = P(A) P(B) P(C)
= 13/52 . 13/52 . 13/52 = 1/4 . 1/4 . 1/4 = 1/64 = 0.016
Question 6.
In Example 6, find (a) the probability that a non-defective part “passes” and (b) the probability that a defective part “fails.”
Answer:
(a) the probability that a non-defective part “passes”
P(P/D) = 3/39 = 1/13 = 0.077
(b) the probability that a defective part “fails.”
P(F/N) = 11/461 = 0.024
Question 7.
At a coffee shop. 80% of customers order coffee. Only 15% of customers order coffee and a bagel. What is the probability that a customer who orders coffee also orders a bagel?
Answer:
A: Customer order coffee
B: Customer order a bagel
P(B/A) = P(A and B)/P(A)
80% of customers order coffee and Only 15% of customers order coffee and a bagel.
P(A) = 80/100 = 0.8
P(A and B) = 15/100 = 0.15
P(B/A) = P(A and B)/P(A) = 0.15/0.8 = 0.1875 = 18.75%
Exercise 12.2 Independent and Dependent Events
Vocabulary and Core Concept Check
Question 1.
WRITING
Explain the difference between dependent events and independent events, and give an example of each.
Answer:
When two events are dependent, the occurrence of one event affects the other. When two events are independent, the occurence of one event does not affect the other.
Question 2.
COMPLETE THE SENTENCE
The probability that event B will occur given that event A has occurred is called the _____________ of B given A and is written as _____________ .
Answer:
The probability that event B will occur given that event A has occurred is called the conditional probability of B given A and is written as P(B/A).
Monitoring Progress and Modeling with Mathematics
In Exercises 3 – 6, tell whether the events are independent or dependent. Explain your reasoning.
Question 3.
A box of granola bars contains an assortment of flavors. You randomly choose a granola bar and eat it. Then you randomly choose another bar.
Event A: You choose a coconut almond bar first.
Event B: You choose a cranberry almond bar second.
Answer:
The two events, which considered in this experiment are an example of dependent events.
Question 4.
You roll a six-sided die and flip a coin.
Event A: You get a 4 when rolling the die.
Event B: You get tails when flipping the coin
Answer: Independent, the events do not influence each other.
Question 5.
Your MP3 player contains hip-hop and rock songs. You randomly choose a song. Then you randomly choose another song without repeating song choices.
Event A: You choose a hip-hop song first.
Event B: You choose a rock song second.
Answer:
The events are dependent because the occurrence of event A affects the occurrence of event B.
Question 6.
There are 22 novels of various genres on a shell. You randomly choose a novel and put it back. Then you randomly choose another novel.
Event A: You choose a mystery novel.
Event B: You choose a science fiction novel.
Answer:
The 1st book chosen is put back so the second book picked has the same probability of being chosen a if the 1st book was never chosen to begin with the events are independent.
In Exercises 7 – 10. determine whether the events are independent.
Question 7.
You play a game that involves spinning a wheel. Each section of the wheel shown has the same area. Use a sample space to determine whether randomly spinning blue and then green are independent events.
Answer:
Question 8.
You have one red apple and three green apples in a bowl. You randomly select one apple to eat now and another apple for your lunch. Use a sample space to determine whether randomly selecting a green apple first and randomly selecting a green apple second are independent events.
Answer:
Let R represent the red apple.
Let G1, G2, G3 represent the 3 green apples.
P(G first) = P(green apple first) = 9/12 = 3/4 = 0.75
P(G second) = P(green apple second) = 9/12 = 3/4 = 0.75
P(green apple first and second) = 6/12 = 1/2 = 0.5
Events are not independent.
Question 9.
A student is taking a multiple-choice test where each question has four choices. The student randomly guesses the answers to the five-question test. Use a sample space to determine whether guessing Question 1 correctly and Question 2 correctly are independent events.
Answer:
Question 10.
A vase contains four white roses and one red rose. You randomly select two roses to take home. Use a sample space to determine whether randomly selecting a white rose first and randomly selecting a white rose second are independent events.
Answer:
P(A and B) = P(A) and P(B)
A =
B =
P(A) = Number of favorable outcomes/Total number of outcomes = 4/5
P(B) = 4/5
P(A) . P(B) = 4/5 . 4/5 = 16/25
Question 11.
PROBLEM SOLVING
You play a game that involves spinning the money wheel shown. You spin the wheel twice. Find the probability that you get more than $500 on your first spin and then go bankrupt on your second spin.
Answer:
Question 12.
PROBLEM SOLVING
You play a game that involves drawing two numbers from a hat. There are 25 pieces of paper numbered from 1 to 25 in the hat. Each number is replaced after it is drawn. Find the probability that you will draw the 3 on your first draw and a number greater than 10 on your second draw.
Answer:
P(A) = 1/25
P(B) = 15/25
P(A and B) = P(A) . P(B)
= 1/25 . 15/25 = 3/125
Thus P(A and B) = 3/125
Question 13.
PROBLEM SOLVING
A drawer contains 12 white socks and 8 black socks. You randomly choose 1 sock and do not replace it. Then you randomly choose another sock. Find the probability that both events A and B will occur.
Event A: The first sock is white.
Event B: The second sock is white.
Answer:
Question 14.
PROBLEM SOLVING
A word game has 100 tiles. 98 of which are letters and 2 of which are blank. The numbers of tiles of each letter are shown. You randomly draw 1 tile, set it aside, and then randomly draw another tile. Find the probability that both events A and B will occur.
Answer:
P(A) = 56/100 = 0.56
P(B) = 42/(100 – 1) = 0.424
P(A) . P(B) = 0.56 × 0.424 = 0.2376
Question 15.
ERROR ANALYSIS
Events A and B are independent. Describe and correct the error in finding P(A and B).
Answer:
Question 16.
ERROR ANALYSIS
A shelf contains 3 fashion magazines and 4 health magazines. You randomly choose one to read, set it aside, and randomly choose another for your friend to read. Describe and correct the error in finding the probability that both events A and B occur.
Event A: The first magazine is fashion.
Event B: The second magazine is health.
Answer:
P(A) = 3/7
P(B/A) = 4/(7 – 1) = 4/6
P(A and B) = P(A) × P(B/A)
P(A and B) = 3/7 × 4/6 = 2/7
Question 17.
NUMBER SENSE
Events A and B are independent. Suppose P(B) = 0.4 and P(A and B) = 0.13. Find P(A).
Answer:
Question 18.
NUMBER SENSE
Events A and B are dependent. Suppose P(B/A) = 0.6 and P(A and B) = 0.15. Find P(A).
Answer:
P(A) = x
P(B/A) = 0.6
P(A and B) = 0.15
P(A and B) = P(A) × P(B/A)
0.15 = x × 0.6
x = 0.15/0.6
x = 0.25
Question 19.
ANALYZING RELATIONSHIPS
You randomly select three cards from a standard deck of 52 playing cards. What is the probability that all three cards are face cards when (a) you replace each card before selecting the next card, and (b) you do not replace each card before selecting the next card? Compare the probabilities.
Answer:
Question 20.
A bag contains 9 red marbles. 4 blue marbles, and 7 yellow marbles. You randomly select three marbles from the hag. what is the probability that all three marbles are red when (a) you replace each marble before selecting the next marble, and (b) you do not replace each marble before selecting the next marble? Compare the probabilities.
Answer:
a. There are a total of 9 + 4 + 7= 20 marbles.
Therefore, the probability of selecting a red marble in each attempt is 9/20 when the marble is replaced.
Therefore the probability of selecting a red marble in each of 3 of the attempt is
9/20 × 9/20 × 9/20 = 0.09125
The replacement makes these independent events.
b. There are total of 9 + 4 + 7 = 20 marbles.
The probability of selecting a red marble in the first attempt is 9/20, second attempt is 8/19 and the third attempt is 7/18 when the marbles are not replaced.
Therefore the probability of selecting a red marble in each of 3 of the attempts is 9/20 × 8/19 × 7/18 = 0.0737
Question 21.
ATTEND TO PRECISION
The table shows the number of species in the United States listed as endangered and threatened. Find (a) the probability that a randomly selected endangered species is a bird, and (b) the probability that a randomly selected mammal is endangered.
Answer:
Question 22.
ATTEND TO PRECISION
The table shows the number of tropical cyclones that formed during the hurricane seasons over a 12-year period. Find (a) the probability to predict whether a Future tropical cyclone in the Northern Hemisphere is a hurricane, and (b) the probability to predict whether a hurricane is in the Southern Hemisphere.
Answer:
Question 23.
PROBLEM SOLVING
At a school, 43% of students attend the homecoming football game. Only 23% of students go to the game and the homecoming dance. What is the probability that a student who attends the football game also attends the dance?
Answer:
Question 24.
PROBLEM SOLVING
At a gas station. 84% of customers buy gasoline. Only 5% of customers buy gasoline and a beverage. What is the probability that a customer who buys gasoline also buys a beverage?
Answer:
Given,
P(A) = 84% = 0.84
P(A and B) = 5% = 0.05
P(A and B) = P(A) × P(B/A)
P(B/A) = 0.05/0.84
P(B/A) = 0.0595
Question 25.
PROBLEM SOLVING
You and 19 other students volunteer to present the “Best Teacher” award at a school banquet. One student volunteer will be chosen to present the award. Each student worked at least 1 hour in preparation for the banquet. You worked for 4 hours, and the group worked a combined total of 45 hours. For each situation, describe a process that gives you a “fair” chance to be chosen. and find the probability that you are chosen.
a. “Fair” means equally likely.
b. “Fair” means proportional to the number of hours each student worked in preparation.
Answer:
Question 26.
HOW DO YOU SEE IT?
A bag contains one red marble and one blue marble. The diagrams show the possible outcomes of randomly choosing two marbles using different methods. For each method. determine whether the marbles were selected with or without replacement.
a.
Answer:
b.
Answer:
Question 27.
MAKING AN ARGUMENT
A meteorologist claims that there is a 70% chance of rain. When it rains. there is a 75% chance that your softball game will be rescheduled. Your friend believes the game is more likely to be rescheduled than played. Is your friend correct? Explain your reasoning.
Answer:
The chance that the game will be rescheduled is (0.7)(0.75) = 0.525
which is 52.5 percent
making it greater than 50 percent.
Question 28.
THOUGHT PROVOKING
Two six-sided dice are rolled once. Events A and B are represented by the diagram. Describe each event. Are the two events dependent or independent? Justify your reasoning.
Answer:
Question 29.
MODELING WITH MATHEMATICS
A football team is losing by 14 points near the end of a game. The team scores two touchdowns (worth 6 points each) before the end of the game. After each touchdown, the coach must decide whether to go for 1 point with a kick (which is successful 99% of the time) or 2 points with a run or pass (which is successful 45% of the time).
a. If the team goes for 1 point after each touchdown, what is the probability that the team wins? loses? ties?
b. If the team goes for 2 points after each touchdown. what is the probability that the team wins? loses? ties?
c. Can you develop a strategy so that the coach’s team has a probability of winning the game that is greater than the probability of losing? If so, explain your strategy and calculate the probabilities of winning and losing the game.
Answer:
Question 30.
ABSTRACT REASONING
Assume that A and B are independent events.
a. Explain why P(B) = P(B/A) and P(A) = P(A/B).
Answer:
P(B) = P(B/A)
P(B) = P(A) . P(B/A)
P(A) = P(A/B).
P(A) = P(B) . P(A/B)
b. Can P(A and B) also be defined as P(B) • P(A/B)? Justify your reasoning.
Answer:
Maintaining Mathematical Proficiency
Solve the equation. Check your solution.
Question 31.
(frac<9><10>) x = 0.18
Answer:
Question 32.
(frac<1><4>)x + 0.5x = 1.5
Answer:
Given,
(frac<1><4>)x + 0.5x = 1.5
0.25x + 0.50x = 1.5
0.75x = 1.5
x = 1.5/0.75
x = 2
Question 33.
0.3x – (frac<3><5>)x + 1.6 = 1.555
Answer:
12.3 Two-Way Tables and Probability
Exploration 1
Completing and Using a Two-Way Table
Work with a partner: A two-way table displays the same information as a Venn diagram. In a two-way table, one category is represented by the rows and the other category is represented by the columns.
The Venn diagram shows the results of a survey in which 80 students were asked whether they play a musical instrument and whether they speak a foreign language. Use the Venn diagram to complete the two-way table. Then use the two-way table to answer each question.
a. How many students play an instrument?
Answer: 41
b. How many students speak a foreign language?
Answer: 46
c. How many students play an instrument and speak a foreign language?
Answer: 16
d. How many students do not play an instrument and do not speak a foreign language?
Answer: 9
e. How many students play an instrument and do not speak a foreign language?
Answer: 25
Exploration 2
Two – Way Tables and Probability
Work with a partner. In Exploration 1, one student is selected at random from the 80 students who took the survey. Find the probability that the student
a. plays an instrument.
Answer: 41/80
b. speaks a foreign language.
Answer: 46/80
c. plays an instrument and speaks a foreign language.
Answer: 16/80
d. does not play an instrument and does not speak a foreign language.
Answer: 9/80
e. plays an instrument and does not speak a foreign language.
Answer:
Exploration 3
Work with your class. Conduct a survey of the students in your class. Choose two categories that are different from those given in Explorations 1 and 2. Then summarize the results in both a Venn diagram and a two-way table. Discuss the results.
MODELING WITH MATHEMATICS
To be proficient in math, you need to identify important quantities in a practical situation and map their relationships using such tools as diagrams and two-way tables.
Answer:
Question 4.
How can you construct and interpret a two-way table?
Answer:
Identify the variables. There are two variables of interest here: the commercial viewed and opinion.
Determine the possible values of each variable. For the two variables, we can identify the following possible values
Set up the table
Fill in the frequencies
Question 5.
How can you use a two-way table to determine probabilities?
Answer:
Lesson 12.3 Two-Way Tables and Probability
Monitoring Progress
Question 1.
You randomly survey students about whether they are in favor of planting a community garden at school. of 96 boys surveyed, 61 are in favor. 0f 88 girls surveyed, 17 are against. Organize the results in a two-way table. Then find and interpret the marginal frequencies.
Answer:
In order to find out how many boys are against you do 96 – 61. In order to find out how many girls are in favor you do 88 – 17.
In order to find the probability you make a proportion. It will be:
P/100 = number of girls against/total number of students
17/184 = p/100
17 × 100 = 184p
1700 = 184p
p = 9.23
Question 2.
Use the survey results in Monitoring Progress Question 1 to make a two-way table that shows the joint and marginal relative frequencies.
Answer:
Question 3.
Use the survey results in Example 1 to make a two-way table that shows the conditional relative frequencies based on the column totals. Interpret the conditional relative frequencies in the context of the problem.
Answer:
Question 4.
Use the survey results in Monitoring Progress Question 1 to make a two-way table that shows the conditional relative frequencies based on the row totals. Interpret the conditional relative frequencies in the context of the problem.
Answer:
Question 5.
In Example 4, what is the probability that a randomly selected customer who is located in Santa Monica will not recommend the provider to a friend?
Answer:
Question 6.
In Example 4, determine whether recommending the provider to a friend and living in Santa Monica are independent events. Explain your reasoning.
Answer:
Question 7.
A manager is assessing three employees in order to offer one of them a promotion. Over a period of time, the manager records whether the employees meet or exceed expectations on their assigned tasks. The table shows the managers results. Which employee should be offered the promotion? Explain.
Answer:
Exercise 12.3 Two-Way Tables and Probability
Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
A(n) ______________ displays data collected from the same source that belongs to two different categories.
Answer:
A two-way table displays data collected from the same source that belongs to two different categories.
Question 2.
WRITING
Compare the definitions of joint relative frequency, marginal relative frequency, and conditional relative frequency.
Answer:
Joint relative frequency: joint relative frequency is the ratio of a frequency that is not in the total row or the total column to the total number of values.
Marginal relative frequency: marginal relative frequency is the sum of the joint relative frequencies in a given row or column.
Conditional relative frequency: It is the ratio of joint relative frequency to the marginal relative frequency.
Monitoring Progress and Modeling with Mathematics
In Exercises 3 and 4, complete the two-way table.
Question 3.
Answer:
Number of students who have passed the exam is 50 – 10 = 40
Of those 40 students, 6 did not study for the exam,
Number of the students who studied and have passed the exam is 40 – 6 = 34
Number of the students who did not study and did not pass the exam is 10 – 4 = 6
Question 4.
Answer:
Number of students who said no is 49 – 7 = 42
Total number of students is 56 + 42 = 98
Total number of people is 98 + 10 = 108
Out of the total number of people, 49 of them said no,
Total number of people who said yes is 108 – 49 = 59
Number of teachers who said yes is 59 – 56 = 3
Question 5.
MODELING WITH MATHEMATICS
You survey 171 males and 180 females at Grand Central Station in New York City. Of those, 132 males and 151 females wash their hands after using the public rest rooms. Organize these results in a two-way table. Then find and interpret the marginal frequencies.
Answer:
Question 6.
MODELING WITH MATHEMATICS
A survey asks 60 teachers and 48 parents whether school uniforms reduce distractions in school. Of those, 49 teachers and 18 parents say uniforms reduce distractions in school, Organize these results in a two-way table. Then find and interpret the marginal frequencies.
Answer:
Given,
A survey asks 60 teachers and 48 parents whether school uniforms reduce distractions in school. Of those, 49 teachers and 18 parents say uniforms reduce distractions in school
Number of teacher who said no is 60 – 49 = 11
Number of parents who said no = 48 – 18 = 30
Total number of people who said yes = 49 + 18 = 67
Total number of people who said no = 11 + 30 = 41
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USING STRUCTURE
In Exercises 7 and 8, use the two-way table to create a two-way table that shows the joint and marginal relative frequencies.
Question 7.
Answer:
P_{1,1} = Number of favorable outcomes/Total number of outcomes
= 11/231 = 0.0476
P_{2,1} = 24/231 = 0.1039
P_{1,2} = 104/231 = 0.4502
P_{2,2} = 92/231 = 0.3983
The marginal relative frequencies we find as the sum of each row and each column.
P(A randomly chosen person is male) = 115/231 = 0.4978
P(A randomly chosen person is female) = 116/231 = 0.5022
P(A randomly chosen person have left dominant hand) = 35/231 = 0.1515
P(A randomly chosen person have left dominant hand) = 196/231 = 0.8484
Question 8.
Answer:
P_{1,1} = Number of favorable outcomes/Total number of outcomes
= 62/410 = 0.1513
The marginal relative frequencies we find as the sum of each row and each column.
P(A randomly chosen person is male) = 377/410 = 0.9195
P(A randomly chosen person is female) = 33/410 = 0.0805
Question 9.
MODELING WITH MATHEMATICS
Use the survey results from Exercise 5 to make a two-way table that shows the joint and marginal relative frequencies.
Answer:
Question 10.
MODELING WITH MATHEMATICS
In a survey, 49 people received a flu vaccine before the flu season and 63 people did not receive the vaccine. Of those who receive the flu vaccine, 16 people got the flu. Of those who did not receive the vaccine, 17 got the flu. Make a two-way table that shows the joint and marginal relative frequencies.
Answer:
We see that the total no. of people who received the vaccine is 49, of which 16 got a fly.
Number of people who received the vaccine and did not get a fly is 49 – 16 = 33
Number of people who did not received the vaccine and did not get a fly is 63 – 17 = 46
Total number of people who got a fly is 16 + 17 = 33
Total Number of people who did not get a fly is 33 + 46 = 79
We also know that the total number of people who were surveyed is 49 + 63 = 112
P_{1,1} = Number of favorable outcomes/Total number of outcomes
= 16/112 = 0.1428
The marginal relative frequencies we find as the sum of each row and each column.
P(A randomly chosen person got a fly) = 33/112 = 0.2946
P(A randomly chosen person did not get a fly) = 79/112 = 0.7053
Question 11.
MODELING WITH MATHEMATICS
A survey finds that 110 people ate breakfast and 30 people skipped breakfast. Of those who ate breakfast. 10 people felt tired. Of those who skipped breakfast. 10 people felt tired. Make a two-way table that shows the conditional relative frequencies based on the breakfast totals.
Answer:
Given,
A survey finds that 110 people ate breakfast and 30 people skipped breakfast.
Of those who ate breakfast. 10 people felt tired. Of those who skipped breakfast. 10 people felt tired.
Number of people who ate breakfast and not tired is 110 – 10 = 100
Number of people who did not eat breakfast and not tired is 30 – 10 = 20
Total number of people who felt tired is 10 + 10 = 20
Total number of people who did not get tired is 100 + 20 = 120
120 +20 = 140
Question 12.
MODELING WITH MATHEMATICS
Use the survey results from Exercise 10 to make a two-way table that shows the conditional relative frequencies based on the flu vaccine totals.
Answer:
Question 13.
PROBLEM SOLVING
Three different local hospitals in New York surveyed their patients. The survey asked whether the patients physician communicated efficiently. The results, given as joint relative frequencies. are shown in the two-way table.
a. What is the probability that a randomly selected patient located in Saratoga was satisfied with the communication of the physician?
b. What is the probability that a randomly selected patient who was not satisfied with the physician’s communication is located in Glens Falls?
c. Determine whether being satisfied with the Communication of the physician and living in Saratoga are independent events.
Answer:
Question 14.
PROBLEM SOLVING
A researcher surveys a random sample of high school students in seven states. The survey asks whether students plan to stay in their home state after graduation. The results, given as joint relative frequencies, are shown in the two-way table.
a. What is the probability that a randomly selected student who lives in Nebraska plans to stay in his or her home state after graduation?
Answer:
In this case we consider a event A =
B =
P(A) = 0.044 + 0.4 = 0.444
P(B/A) = P(A and B)/P(A) = 0.044/0.444 = 0.099
About 1% students who lives Nebraska plans to stay in his or her home state after graduation.
b. What is the probability that a randomly selected student who does not plan to stay in his or her home state after graduation lives in North Carolina?
Answer:
C =
D =
P(C) = 0.4 + 0.193 + 0.256 = 0.849
P(D/C) = P(C and D)/P(C)
= 0.193/0.849 = 0.227
Therefore about 22.7% students who does not plan to stay in his or her home state after graduation lives in North Carolina.
ERROR ANALYSIS
In Exercises 15 and 16, describe and correct the error in finding the given conditional probability.
Question 15.
P(yes|Tokyo)
Answer:
Question 16.
P(London|No)
Answer:
P(A and B) = P(A)P(B/A)
P(A) = 0.341 + 0.112+ 0.191 = 0.644
P(B/A) = P(A and B)/P(A) = 0.112/0.644 = 0.1739
In the denominator the probability P(B) = 0.248 is used instead of P(A), where P(B) is probability that a randomly chosen person live in London.
Question 17.
PROBLEM SOLVING
You want to find the quickest route to school. You map out three routes. Before school, you randomly select a route and record whether you are late or on time. The table shows your findings. Assuming you leave at the same time each morning, which route should you use? Explain.
Answer:
Question 18.
PROBLEM SOLVING
A teacher is assessing three groups of students in order to offer one group a prize. Over a period of time, the teacher records whether the groups meet or exceed expectations on their assigned tasks. The table shows the teacher’s results. Which group should be awarded the prize? Explain.
Answer: Group 1 exceeded expectations 12 out of 16 times or 75% of the time.
Question 19.
OPEN-ENDED
Create and conduct a survey in your class. Organize the results in a two-way table. Then create a two-way table that shows the joint and marginal frequencies.
Answer:
Question 20.
HOW DO YOU SEE IT?
A research group surveys parents and Coaches of high school students about whether competitive sports are important in school. The two-way table shows the results of the survey.
a. What does 120 represent?
Answer: 120 parents said that competitive sports are not important in school.
b. What does 1336 represent?
Answer: 1336 is the sum of parents and coaches who agree that competitive sports are important in school.
c. What does 1501 represent?
Answer: 1501 is the total number of people surveyed. Here this is the sum of the parents and the coaches who participated in the survey.
Question 21.
MAKING AN ARGUMENT
Your friend uses the table below to determine which workout routine is the best. Your friend decides that Routine B is the best option because it has the fewest tally marks in the “Docs Not Reach Goal” column. Is your friend correct? Explain your reasoning.
Answer:
Question 22.
MODELING WITH MATHEMATICS
A survey asks students whether they prefer math class or science class. Of the 150 male students surveyed, 62% prefer math class over science class. Of the female students surveyed, 74% prefer math. Construct a two-way table to show the number of students in each category if 350 students were surveyed.
Answer:
survey asks students whether they prefer math class or science class. Of the 150 male students surveyed, 62% prefer math class over science class.
62% = 0.62
0.62 = P(Math/Male)
= P(Math and Male)/P(Male)
= Number of male students who prefer math/150
Number of male students who prefer math = 0.62 × 150 = 93
0.74 = P(Math/Female)
= P(Math and Female)/P(Female)
= Number of female students who prefer math/200
Number of female students who prefer math = 0.74 × 200 = 148
So, the total number of students who prefer math class is 148 + 93 = 241
Number of male students who prefer science -93 = 57
Number of female students who prefer science – 148 = 52
Number of students who prefer science + 52 = 109
Question 23.
MULTIPLE REPRESENTATIONS
Use the Venn diagram to construct a two-way table. Then use your table to answer the questions.
a. What is the probability that a randomly selected person does not own either pet?
b. What is the probability that a randomly selected person who owns a dog also owns a cat?
Answer:
Question 24.
WRITING
Compare two-way tables and Venn diagrams. Then describe the advantages and disadvantages of each.
Answer:
Question 25.
PROBLEM SOLVING
A company creates a new snack, N, and tests it against its current leader, L. The table shows the results.
The company is deciding whether it should try to improve the snack before marketing it, and to whom the snack should be marketed. Use probability to explain the decisions the company should make when the total size of the snack’s market is expected to (a) change very little, and (b) expand very rapidly.
Answer:
Question 26.
THOUGHT PROVOKING
Baye’s Theorem is given by
Use a two-way table to write an example of Baye’s Theorem.
Answer:
P(Cat owner) = 61/210 = 0.29
P(Dog owner) = 93/210 = 0.442
P(Cat Owner/Dog owner) = P(Dog owner and cat owner)/P(Dog owner) = 0.387
P(Dog owner/Cat owner) = P(Cat owner/Dog owner)P(Dog owner)/P(Cat owner)
= 0.387 × 0.442/0.29
= 0.5898
Maintaining Mathematical Proficiency
Draw a Venn diagram of the sets described.
Question 27.
Of the positive integers less than 15, set A consists of the factors of 15 and set B consists of all odd numbers.
Answer:
Question 28.
Of the positive integers less than 14, set A consists of all prime numbers and set B consists of all even numbers.
Answer:
Set A = <2. 3, 5, 7, 11, 13>
Set B = <2, 4, 6, 8, 10, 12>
It can be seen that here A and B are overlapping sets.
Question 29.
Of the positive integers less than 24, set A consists of the multiples of 2 and set B consists of all the multiples of 3.
Answer:
12.1 – 12.3 Quiz
Question 1.
You randomly draw a marble out of a bag containing 8 green marbles, 4 blue marbles 12 yellow marbles, and 10 red marbles. Find the probability of drawing a marble that is not yellow.
Answer:
Given,
You randomly draw a marble out of a bag containing 8 green marbles, 4 blue marbles 12 yellow marbles, and 10 red marbles.
Total number of outcomes here are 8 + 4 + 12 + 10 = 34
Thus the probability of obtaining a marble that is not yellow is (8 + 4 + 10)/34
= 22/34
= 11/17
= 0.647
= 64.7%
Thus the probability of obtaining a marble that is not yellow is 64.7%
Question 5.
You roll a six-sided die 30 times. A 5 is rolled 8 times. What is the theoretical probability of rolling a 5? What is the experimental probability of rolling a 5?
Answer:
Given,
You roll a six-sided die 30 times. A 5 is rolled 8 times.
The theoretical probability of rolling a 5 on a number cube is (frac<1><6>) while the experimental probability of rolling a 5 on a number cube is (frac<8><30>) = (frac<4><15>)
Question 6.
Events A and B are independent. Find the missing probability.
P(A) = 0.25
P(B) = ____
P(A and B) = 0.05
Answer:
Given,
P(A) = 0.25
P(A and B) = 0.05
P(A and B) = P(A) × P(B)
P(B) = P(A and B)/P(A)
P(B) = 0.05/0.25 = 0.2
P(B) = 0.2
Question 7.
Events A and B are dependent. Find the missing probability.
P(A) = 0.6
P(B/A) = 0.2
P(A and B) = ____
Answer:
Given,
P(A) = 0.6
P(B/A) = 0.2
P(A and B) = P(A) × P(B/A)
P(A and B) = 0.6 × 0.2
= 0.12
P(A and B) = 0.12
Question 8.
Find the probability that a dart thrown at the circular target Shown will hit the given region.
Assume the dart is equally likely to hit any point inside the target.
a. the center circle
Answer:
Total area of the given region is π × r² = π × 6² = 36π = 113.112 sq. units
Area of the center circle is π × r² = π × 2² = 4π sq. units.
Therefore the probability of hitting the center circle is 4π/36π = 1/9 = 0.11…
b. outside the square
Answer:
Area of the square is 6² = 36
So the region outside of it is equal to 36π – 36 = 77.112 sq. units
Thus the probability of hitting the region outside the square is 77.112/113.112 = 0.682
c. inside the square but outside the center circle
Answer:
Area of the center circle is π × r² = π × 2² = 4π sq. units.
Area of the square is 6² = 36
Thus the probability of hitting the region outside the center circle but inside the square is 36 – 4π = 23.432 sq. units
Thus the probability of hitting the region is 23.432/113.112 = 0.207
Question 9.
A survey asks 13-year-old and 15-year-old students about their eating habits. Four hundred students are surveyed, 100 male students and 100 female students from each age group. The bar graph shows the number of students who said they eat fruit every day.
a. Find the probability that a female student, chosen at random from the students surveyed, eats fruit every day.
Answer:
Total number of females who eat a fruit everyday are 61 + 58 = 119
Therefore the probability of randomly choosing a female who eats a fruit everyday is 119/400= 0.2975
b. Find the probability that a 15 – year – old student. chosen at random from the students surveyed, eats fruit every day.
Answer:
Total number of 15 year old student who eat a fruit everyday are 53 + 58 = 111
Therefore the probability of randomly choosing a 15 year old student who eats a fruit everyday is 111/200 = 0.555
Question 10.
There are 14 boys and 18 girls in a class. The teacher allows the students to vote whether they want to take a test on Friday or on Monday. A total of 6 boys and 10 girls vote to take the test on Friday. Organize the information in a two-way table. Then find and interpret the marginal frequencies.
Answer:
Given,
There are 14 boys and 18 girls in a class. The teacher allows the students to vote whether they want to take a test on Friday or on Monday.
14 + 18 = 32
Number of boys who vote to take the test on Monday is 14 – 6 = 8
Number of girls who vote to take the test on monday is 18 – 10 = 8
A total of 6 boys and 10 girls vote to take the test on Friday.
The total number of students who take the test on Friday is 10 + 6 = 16
The total number of students who vote to take the test on Friday is 8 + 8 = 16
Question 11.
Three schools compete in a cross country invitational. Of the 15 athletes on your team. 9 achieve their goal times. Of the 20 athletes on the home team. 6 achieve their goal times. On your rival’s team, 8 of the 13 athletes achieve their goal times. Organize the information in a two-way table. Then determine the probability that a randomly elected runner who achieves his or her goal time is from your school.
Answer:
Three schools compete in a cross country invitational. Of the 15 athletes on your team. 9 achieve their goal times.
Number of runners in your team who do not achieve their goal team is 15 – 9 = 6
Number of runners in home team who do not achieve their goal team is 20 – 6 = 14
Number of runners in rival’s team who do not achieve their goal team is 13 – 8 = 5
Total number of runners who achieve their goal team is 9 + 6 + 8 = 23
Total number of runners who do not achieve their goal team is 6 + 14 + 5 = 25
The total number of rubbers who was surveyed is 23 + 25 = 48
P = Your team ans achieve their goal team/P(Archive their goal team)
P = 9/23
P = 0.39
12.4 Probability of Disjoint and Overlapping Events
Exploration 1
Work with a partner: A six-sided die is rolled. Draw a Venn diagram that relates the two events. Then decide whether the cents are disjoint or overlapping.
MODELING WITH MATHEMATICS
To be proficient in math, you need to map the relationships between important quantities in a practical situation using such tools as diagrams.
a. Event A: The result is an even number.
Event B: The result is a prime number.
Answer:
b. Event A: The result is 2 or 4.
Event B: The result is an odd number
Answer:
Exploration 2
Finding the Probability that Two Events Occur
Work with a partner: A six-sided die is roiled. For each pair of events. find (a) P(A), (b) P(B). (C) P(A) and (B). and (d) P(A or B).
a. Event A: The result is an even number.
Event B: The result is a Prime number.
Answer:
P(A) = (frac<3><6>) = (frac<1><2>)
P(B) = (frac<3><6>) = (frac<1><2>)
P(A or B) = P(A) + P(B) – P(A and B)
P(A and B) = (frac<1><6>)
P(A or B) = (frac<5><6>)
b. Event A: The result is 2 or 4.
Event B: The result is an odd number.
Answer:
P(A) = (frac<2><6>) = (frac<1><3>)
P(B) = (frac<3><6>) = (frac<1><2>)
P(A or B) = P(A) + P(B) – P(A and B)
P(A and B) = 0
P(A or B) = (frac<5><6>)
Exploration 3
Discovering Probability Formulas
Work with a partner:
a. In general, if event A and event B arc disjoint, then what is the probability that event A or event B will occur? Use a Venn diagram to justify your conclusion.
Answer:
If event A and B are disjoint, there are no common outcomes.
So we add the probabilities that each event occurs:
P(A or B) = P(A) + P(B)
b. In general, if event A and event B are overlapping, then what is the probability that event A or event B will occur? Use a Venn diagram to justify your conclusion.
Answer:
If event A and event B are overlapping, there are common outcomes.
So, we add the probabilities that each event occurs then subtract the probability of the common outcomes.
P(A or B) = P(A) + P(B) – P(A and B)
c. Conduct an experiment using a six-sided die. Roll the die 50 times and record the results. Then use the results to find the probabilities described in Exploration 2. How closely do your experimental probabilities compare to the theoretical probabilities you found in Exploration 2?
Answer:
a. P(A) = (frac<1><2>) = 50%
P(A) = (frac<21><50>) = 42%
P(B) = (frac<1><2>) = 50%
P(B) = (frac<32><50>) = 64%
P(A and B) = (frac<1><6>) ≈ 16.7%
P(A and B) = (frac<9><50>) ≈ 18%
P(A or B) = (frac<5><6>) ≈ 83.3%
P(A or B) = (frac<44><50>) ≈ 88%
P(A) = (frac<1><3>) ≈ 33.3%
P(A) = (frac<17><50>) = 34%
P(B) = (frac<1><2>) = 50%
P(B) = (frac<29><50>) = 58%
P(A and B) = 0 = 0%
P(A and B) = (frac<0><50>) = 0%
P(A or B) = (frac<5><6>) ≈ 83.3%
P(A or B) = (frac<46><50>) = 92%
Question 4.
How can you find probabilities of disjoint and overlapping events?
Answer:
If A and B are disjoint events, then the probability of A or B is P(A or B) = P(A) + P(B). If two events A and B are overlapping, then the outcomes in the intersection of A and B are counted twice when P(A) and P(B) are added.
P(A or B) = P(A) + P(B) – P(A and B)
Question 5.
Give examples of disjoint events and overlapping events that do not involve dice.
Answer:
a. Event A: The result is an even number.
Event B: The result is a prime number.
Answer:
b. Event A: The result is 2 or 4.
Event B: The result is an odd number
Answer:
Lesson 12.4 Probability of Disjoint and Overlapping Events
Monitoring Progress
A card is randomly selected from a standard deck of 52 playing cards. Find the probability of the event.
Question 1.
selecting an ace or an 8
Answer:
A: Selecting an ace
B: You select 8
We know that A has 4 outcomes and B also has 4 outcomes.
P(A or B) = P(A) + P(B)
= 4/52 + 4/52
= 8/52
= 2/13 ≈ 0.15
Question 2.
selecting a 10 or a diamond
Answer:
A: Selecting a 10
B: You select diamond
P(A or B) = P(A) + P(B) – P(A and B)
= 4/52 + 13/52 – 1/52
= 16/52
= 4/13
Question 3.
WHAT IF?
In Example 3, suppose 32 seniors are in the band and 64 seniors are in the band or on the honor roll. What is the Probability that a randomly selected senior is both in the band and on the honor roll?
Answer:
Question 4.
In Example 4, what is the probability that the diagnosis is incorrect?
Answer:
Question 5.
A high school basketball team leads at halftime in 60% of the games in a season. The team wins 80% of the time when the have the halftime lead, but only 10% of the time when the do not. What is the probability that the team wins a particular game during the season?
Answer:
Given,
A high school basketball team leads at halftime in 60% of the games in a season. The team wins 80% of the time when the have the halftime lead, but only 10% of the time when the do not.
Let event A be team leads on the halftime and event B be win.
When A occurs, P(B) = 0.8
When A does not occur, P(B) = 0.1
P(B) = P(A and B) + P((ar) and B)
P(A) . P(B | A) + P((ar)) . P(B | (ar))
= 0.6 × 0.8 + 0.4 × 0.1
= 0.52
Exercise 12.4 Probability of Disjoint and Overlapping Events
Vocabulary and Core Concept Check
Question 2.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.
How many outcomes are in the intersection of A and B?
Answer: There are 2 outcomes in the intersection of A and B.
How many outcomes are shared by both A and B?
Answer: 2 outcomes are shared by both A and B.
How many outcomes are in the union of A and B?
Answer: There are 4 + 2 + 3 = 9 outcomes in the union of A and B.
How many outcomes in B are also in A?
Answer: There are 2 outcomes in B that are also in A.
Monitoring progress and Modeling with Mathematics
In Exercises 3 – 6, events A and B are disjoint. Find P(A or B)
Question 3.
p(A) = 0.3, P(B) = 0.1
Answer:
Question 4.
p(A) = 0.55, P(B) = 0.2
Answer:
Given,
p(A) = 0.55, P(B) = 0.2
P(A or B) = P(A) + P(B)
P(A or B) = 0.55 + 0.2 = 0.75
Question 5.
P(A) = (frac<1><3>), P(B) = (frac<1><4>)
Answer:
Question 7.
PROBLEM SOLVING
Your dart is equally likely to hit any point inside the board Shown. You throw a dart and pop a balloon. What is the probability that the balloon is red or blue?
Answer:
Question 8.
PROBLEM SOLVING
You and your friend are among several candidates running for class president. You estimate that there is a 45% chance you will win and a 25% chance your friend will win. What is the probability that you or your friend win the election?
Answer:
Let A be the event of you winning the election and B be of your friend winning the election
P(A) = 45% = 0.45
P(B) = 25% = 0.25
P(A or B) = P(A) + P(B)
P(A or B) = 0.45 + 0.25 = 0.70
Therefore the probability of you or your friend winning the election is 0.7
Question 9.
PROBLEM SOLVING
You are performing an experiment to determine how well plants grow under different light sources. 0f the 30 Plants in the experiment, 12 receive visible light, 15 receive ultraviolet light, and 6 receive both visible and ultraviolet light. What is the probability that a plant in the experiment receives visible or ultraviolet light?
Answer:
Question 10.
PROBLEM SOLVING
Of 162 students honored at an academic awards banquet, 48 won awards for mathematics and 78 won awards for English. There are 14 students who won awards for both mathematics and English. A newspaper chooses a student at random for an interview. What is the probability that the student interviewed won an award for English or mathematics?
Answer:
Given,
There are 162 students honored at an academic awards banquet, 48 won awards for mathematics and 78 won awards for English.
There are 14 students who won awards for both mathematics and English. A newspaper chooses a student at random for an interview.
P(A) = 48/162
P(B) = 78/162
P(A and B) = 14/162
P(A or B) = P(A) + P(B) – P(A and B)
P(A or B) = 48/162 + 78/162 – 14/162
P(A or B) = 112/162 = 56/81 = 0.691
ERROR ANALYSIS
In Exercises 11 and 12, describe and correct the error in finding the probability of randomly drawing the given card from a standard deck of 52 playing cards.
Question 11.
Answer:
Question 12.
Answer:
These 2 events are overlapping events as there is 1 card that is both a club and 9, therefore write equation of P(A or B) for overlapping events
P(A or B) = P(A) + P(B) – P(A and B)
= 13/52 + 4/52 – 1/52
= 4/13
In Exercises 13 and 14, you roll a six-sided die. Find P(A or B).
Question 13.
Event A: Roll a 6.
Event B: Roll a prime number.
Answer:
Question 14.
Event A: Roll an odd number.
Event B: Roll a number less than 5.
Answer:
P(A) = 3/6
1, 3, 5 out of 6 possible outcomes
P(B) = 4/6
1, 2, 3, 4 out of 6 possible outcomes
P(A and B) = 2/6
3/4 + 4/6 – 2/6 = 5/6
Question 15.
DRAWING CONCLUSIONS
A group of 40 trees in a forest are not growing properly. A botanist determines that 34 of the trees have a disease or are being damaged by insects, with 18 trees having a disease and 20 being damaged by insects. What is the probability that a randomly selected tree has both a disease and is being damaged by insects?
Answer:
Question 16.
DRAWING CONCLUSIONS
A company paid overtime wages or hired temporary help during 9 months of the year. Overtime wages were paid during 7 months. and temporary help was hired during 4 months. At the end of the year, an auditor examines the accounting records and randomly selects one month to check the payroll. What is the probability that the auditor will select a month in which the company paid overtime wages and hired temporary help?
Answer:
P(A) = 7/12
P(B) = 4/12
P(A or B) = 9/12
P(A and B) = P(A) + P(B) – P(A or B)
P(A and B) = 7/12 + 4/12 – 9/12
P(A and B) = 2/12 = 1/6
The probability of randomly selecting a month in which overtime was paid and temporary help was hired is 1/6 = 0.166…
Question 17.
DRAWING CONCLUSIONS
A company is focus testing a new type of fruit drink. The focus group is 47% male. 0f the responses, 40% of the males and 54% of the females said they would buy the fruit drink. What is the probability that a randomly selected person would buy the fruit drink?
Answer:
Question 18.
DRAWING CONCLUSIONS
The Redbirds trail the Bluebirds by one goal with 1 minute left in the hockey game. The Redbirds coach must decide whether to remove the goalie and add a frontline player. The probabilities of each team scoring are shown in the table.
a. Find the probability that the Redbirds score and the Bluebirds do not score when the coach leaves the goalie in.
Answer:
Redbirds 10% x Bluebirds 90% = 9%
b. Find the probability that the Redbirds score and the Bluebirds do not score when the coach takes the goalie out.
Answer:
Redbirds 30% x Bluebirds 40% = 12%
c. Based on parts (a) and (b), what should the coach do?
Answer: Looks to be a 3% better chance to tie it up in B – pull the goalie
Question 19.
PROBLEM SOLVING
You can win concert tickets from a radio station if you are the first person to call when the song of the day is played. or if you are the first person to correctly answer the trivia question. The song of the day is announced at a random time between 7:00 and 7:30 A.M. The trivia question is asked at a random Lime between 7:15 and 7:45 A.M. You begin listening to the radio station at 7:20. Find the probability that you miss the announcement of the song of the day or the trivia question.
Answer:
Question 20.
HOW DO YOU SEE IT?
Are events A and B disjoint events? Explain your reasoning.
Answer:
A and B are not disjoint events and in fact they are overlapping events with 1 overlapping outcome.
Disjoint events do not have any overlap and they are mutually exclusive from one another.
Question 21.
PROBLEM SOLVING
You take a bus from sour neighborhood to your school. The express bus arrives at your neighborhood at a random time between 7:30 and 7:36 AM. The local bus arrives at your neighborhood at a random time between 7:30 and 7:40 A.M. You arrive at the bus stop at 7:33 A.M. Find the probability that you missed both the express bus and the local bus.
Answer:
Question 22.
THOUGHT PROVOKING
Write a general rule for finding P(A or B or C) for (a) disjoint and (b) overlapping events A, B, and C.
Answer:
For 2 disjoint events, the equation becomes:
P(A or B) = P(A) + P(B), based on this it can be said that the equation of P(A or B or C) will be
P(A or B or C) = P(A) + P(B) + P(C)
For 2 overlapping events, the equation becomes:
P(A or B) = P(A) + P(B) – P(A and B)
P(A or B or C) = P(A) + P(B) + P(C) – P(A and B) – P(A and C) – P(B and C) + P(A and B and C)
Question 23.
MAKING AN ARGUMENT
A bag contains 40 cards numbered 1 through 40 that are either red or blue. A card is drawn at random and placed back in the bag. This is done four times. Two red cards are drawn. numbered 31 and 19, and two blue cards are drawn. numbered 22 and 7. Your friend concludes that red cards and even numbers must be mutually exclusive. Is your friend correct? Explain.
Answer:
Your friend is incorrect because we do not know all the number of cards. Also, from the given data we do not know all colors for cards. Therefore we can not conclude that red cards and even numbers be mutually exclusive.
Maintaining Mathematical Proficiency
Question 24.
(n – 12) 2
Answer:
We can solve the product by using the formula
(a – b)² = a² – 2ab + b²
(n – 12) 2 = (n)² – 2(n)(12) + (12)²
n² – 24n + 144
Question 25.
(2x + 9) 2
Answer:
Question 26.
(- 5z + 6) 2
Answer:
(- 5z + 6) 2 = (6 – 5z) 2
We can solve the product by using the formula
(a – b)² = a² + 2ab + b²
(6 – 5z) 2 = (6)² – 2(6)(5z) + (5z)²
36 – 24n + 25z²
Question 27.
(3a – 7b) 2
Answer:
We can solve the product by using the formula
(a – b)² = a² + 2ab + b²
(3a – 7b) 2 = (3a)² – 2(3a)(7b) + (7b)²
9a² – 42ab + 49b²
12.5 Permutations and Combinations
Exploration 1
Work with a partner. Two coins are flipped and the spinner is spun. The tree diagram shows the possible outcomes.
a. How many outcomes are possible?
Answer:
b. List the possible outcomes.
Answer:
Exploration 2
Work with a partner: Consider the tree diagram below.
a. How many events are shown?
Answer:
b. What outcomes are possible for each event?
Answer:
c. How many outcomes are possible?
Answer:
d. List the possible outcomes.
Answer:
Exploration 3
a. Consider the following general problem: Event 1 can occur in in ways and event 2 can occur in n ways. Write a conjecture about the number of ways the two events can occur. Explain your reasoning.
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math,
you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.
Answer:
b. Use the conjecture you wrote in part (a) to write a conjecture about the number of ways more than two events can occur. Explain your reasoning.
Answer:
c. Use the results of Explorations 1(a) and 2(c) to verify your conjectures.
Answer:
Question 4.
How can a tree diagram help you visualize the number of ways in which two or more events can occur?
Answer:
Question 5.
In Exploration 1, the spinner is spun a second time. How many outcomes are possible?
Answer:
Lesson 12.5 Permutations and Combinations
Monitoring Progress
Question 1.
In how many ways can you arrange the letters in the word HOUSE?
Answer:
Question 2.
In how many ways can you arrange 3 of the letters in the word MARCH?
Answer:
Question 3.
WHAT IF
In Example 2, suppose there are 8 horses in the race. In how many different ways can the horses finish first, second, and third? (Assume there are no ties.)
Answer:
Question 4.
WHAT IF?
In Example 3, suppose there are 14 floats in the parade. Find the probability that the soccer team is first and the chorus is second.
Answer:
Question 5.
Count the possible combinations of 3 letters chosen from the list A, B, C, D, E.
Answer:
Question 6.
WHAT IF?
In Example 5, suppose you can choose 3 side dishes out of the list of 8 side dishes. How many combinations are possible?
Answer:
Question 7.
WHAT IF?
In Example 6, suppose there are 20 photos in the collage. Find the probability that your photo and your friend’s photo are the 2 placed at the top of the page.
Answer:
Exercise 12.5 Permutations and Combinations
Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
An arrangement of objects in which order is important is called a(n) _________ .
Answer:
An arrangement of objects in which order is important is called a Permutation.
Question 2.
WHICH ONE DOESN’T BELONG?
Which expression does not belong with the other three? Explain your reasoning.
(frac<7 !><2 ! cdot 5 !>) _{7}C_{5} _{7}C_{2} (frac<7 !><(7-2) !>)
Answer:
_{7}C_{2} (frac<7 !><(7-2) !>) = (frac<7 !><5!>)
= _{7}C_{2}
The expression (frac<7 !><(7-2) !>) does not belong with other three.
Monitoring Progress and Modeling with Mathematics
In Exercises 3 – 8, find the number of ways you can arrange (a) all of the letters and (b) 2 of the letters in the given word.
Question 3.
AT
Answer:
Question 4.
TRY
Answer:
a. In this case, we have to find the number of permutations all of the letters in a given word that will consist of 3 letters.
Number of permutations = (1st place can be one of three letters) × (2nd can be one of two letters that is left) × (3rd can be one letter that is left)
= 3 . 2 . 1 = 6
Therefore we have 6 ways for arrange all of the letters in given word, that is TRY, TYR, YTR, YRT, RTY and RYT.
Now, we have to find the number of permutation 2 of the letters in a given word that will consists of 3 letters
Number of permutations = (1st place can be one of three letters) × (2nd can be one of two letters that is left)
= 3 . 2 = 6
We have 6 ways for arrange 2 of the letters in given word, tthat is TR, TY, YT, YR, RT and RY.
Question 5.
ROCK
Answer:
Question 6.
WATER
Answer:
In this case, we have to find the number of permutation all of the letters in a given word that will consist of 5 letters.
Number of permutations = (1st can be one of 5 letters) × (2nd place can be one of 4 letters that is left) × (3rd can be one of 3 letters that is left) × (4th can be two letters that is left) × (5th can be one letter that is left)
= 5 .4 . 3 . 2 . 1 = 120
Thus we have 120 ways to arrange all of the letters in given word, that WATER, WATRE, WARTE,…, RETAW.
Number of permutations = (1st can be one of 5 letters) × (2nd place can be one of 4 letters that is left)
= 5 . 4
= 20
Thus we have 20 ways to arrange 2 of the letters in given word, that WA, WT, WR, WE …, ER, RE.
Question 7.
FAMILY
Answer:
Question 8.
FLOWERS
Answer:
We have to find the number of permutation all of the letters in a given word that will consist of 7 letters.
Number of permutations = (1st place can be one of 7 letters) × (2nd place can be one of 6 letters that is left) × (3rd place can be one of 5 letters that is left) × (4th place can be one of 4 letters that is left) × (5th place can be one of 3 letters that is left) × (6th place can be one of 2 letters that is left) × (7th place can be one of 1 letters that is left)
= 7 . 6 . 5 . 4 . 3 . 2 . 1 = 5040
Thus we have 5040 ways to arrange all of the letters in given word, that is FLOWERS, FLOWERS, FLOWSER… SREWOLF
Now we have to find the number of permutation 2 of the letters in a given word that will consist of 7 letters.
Number of permutations = (1st place can be one of 7 letters) × (2nd place can be one of 6 letters that is left)
= 7 . 6
= 42
Thus we have 42 ways to arrange all of the letters in given word, that is FL, FO, FW, FE…RS, RE.
In Exercises 9 – 16, evaluate the expression.
Question 9.
_{5}P_{2}
Answer:
Question 11.
_{9}P_{1}
Answer:
Question 13.
_{8}P_{6}
Answer:
Question 15.
_{30}P_{2}
Answer:
Question 16.
_{25}P_{5}
Answer:
_{25}P_{5} = (frac<25 !><(25-5) !>) = (frac<25 !><20!>)
= 25 . 24 . 23 . 22 . 21 . 20 . 19 . 18 . …. 6 . 5 . 4 . 3 . 2 . 1/20 . 19 . 18 . … 3 . 2 . 1
= 25 . 24 . 23 . 22 . 21
= 720
_{25}P_{5} = 6375600
Question 17.
PROBLEM SOLVING
Eleven students are competing in an art contest. In how many different ways can the students finish first, second, and third?
Answer:
Question 18.
PROBLEM SOLVING
Six Friends go to a movie theater. In how many different ways can they sit together in a row of 6 empty seats?
Answer:
Given,
Six Friends go to a movie theater.
_{6}P_{6} = (frac<6!><(6-6) !>)
= (frac<6!><0!>)
= 6!
= 6 . 5 . 4 . 3 . 2 . 1
_{6}P_{6} = 720
Question 19.
PROBLEM SOLVING
You and your friend are 2 of 8 servers working a shill in a restaurant. At the beginning of the shill. the manager randomly assigns 0ne section to each server. Find the probability that you are assigned Section 1 and your friend is assigned Section 2.
Answer:
Question 20.
PROBLEM SOLVING
You make 6 posters to hold up at a basketball game. Each poster has a letter of the word TIGERS. You and 5 friends sit next to each other in a row. The posters are distributed at random. Find the probability that TIGERS is spelled correctly when you hold up the posters.
Answer:
Number of favorable outcomes = 1 (TIGERS)
Total number of outcomes = 6!
= 6 . 5 . 4 . 3 . 2 . 1
= 720
Number of favorable outcomes/Total number of outcomes = 1/720
In Exercises 21 – 24, count the possible combinations of r letters chosen from the given list.
Question 21.
A, B, C, D r = 3
Answer:
Question 22.
L, M, N, O r = 2
Answer:
_{4}P_{2} = (frac<4!><(4-2) !>)
= (frac<4!><2!>)
= 4 . 3 . 2 . 1/2 . 1
= 12
_{4}P_{2} = 12
So, the possible permutations of two letters in the given list L, M, N, O is
LM, ML
LN, NL
LO, OL
MN, NM
MO, MO
NO, ON
Thus the number of possible combination of a = 2 letters chosen from the list L, M, N, O
_{4}P_{2} = 12/2 = 6
Question 23.
U , V, W, X, Y, Z r = 3
Answer:
Question 24.
D, E, F, G, H R = 4
Answer:
_{5}P_{4} = (frac<5!><(5-4) !>)
= (frac<5!><1!>)
=5 . 4 . 3 . 2 . 1
= 120
_{5}P_{4} = 120
Thus the possible permutations of four letters in the given list D, E, F, G, H is
DEFG, DEGF, DGEF, DGFE, DFGD…HEFG, EHFG, EHGF, FGEH.
Thus we conclude that the number of possible combination of a = 4 letters chosen from the list D, E, F, G, H is
_{5 C} _{4} = (frac<120><(24) !>) = 5
In Exercise 25 – 32, evaluate the expression
Question 25.
_{5}C_{1}
Answer:
Question 27.
_{9}C_{9}
Answer:
Question 29.
_{12}C_{3}
Answer:
Question 30.
_{11}C_{4}
Answer:
_{11 C} _{4} = (frac<11!><(11-4) !>)
= (frac<11!><7!>) . (frac<1><4!>)
= 11 . 10 . 9 . 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1/(7 . 6. 5 . 4 . 3 . 2 . 1) (4 . 3 . 2 . 1)
= 11 . 10 . 3
_{11 C} _{4} = 330
Question 31.
_{15}C_{8}
Answer:
Question 32.
_{20}C_{5}
Answer:
_{20 C} _{5} = (frac<20!><(20-5) !>)
= (frac<20!><5!>) . (frac<1><5!>)
= 20 . 19 . 18 . 17 . 16 . 15 . … 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1/(5 . 4 . 3 . 2 . 1) (5 . 4 . 3 . 2 . 1)
= 19 . 3 . 17 . 16
_{20 C} _{5} = 15504
Question 33.
PROBLEM SOLVING
Each year, 64 golfers participate in a golf tournament. The golfers play in groups of 4. How many groups of 4 golfers are possible?
Answer:
Question 34.
PROBLEM SOLVING
You want to purchase vegetable dip for a party. A grocery store sells 7 different flavors of vegetable dip. You have enough money to purchase 2 flavors. How many combinations of 2 flavors of vegetable dip are possible?
Answer:
given that,
You want to purchase vegetable dip for a party. A grocery store sells 7 different flavors of vegetable dip. You have enough money to purchase 2 flavors.
_{7 C} _{2} = (frac<7!><(7-2) !>)
= (frac<7!><5!>) . (frac<1><2!>)
= 7 . 6 . 5 . 4 . 3 . 2 . 1/(5 . 4 . 3 . 2 . 1) (2 . 1)
= 7 . 3
_{7 C} _{2} = 21
ERROR ANALYSIS
In Exercises 35 and 36, describe and correct the error in evaluating the expression.
Question 35.
Answer:
Question 36.
Answer:
The permutation formula was used instead of the combination formula.
REASONING
In Exercises 37 – 40, tell whether the question can be answered using permutations or combinations. Explain your reasoning. Then answer the question.
Question 37.
To complete an exam. u must answer 8 questions from a list of 10 questions. In how many ways can you complete the exam?
Answer:
Question 38.
Ten students are auditioning for 3 different roles in a play. In how many ways can the 3 roles be filled?
Answer:
As 10 students are auditioning for 3 roles, which are different from each other, the order in which the roles are assigned to the students is important and should be taken into account.
As the Permutations formula takes into account the order of distribution. Hence the number of ways to fill the 3 roles can be found by using the permutations formula in the chapter.
So, the number of permutations of assigning the 3 roles to 3 students chosen from 10, using the permutations formula
_{10 P} _{3} = (frac<10!><(10-3) !>)
= (frac<10!><7!>)
= 10 . 9 . 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1/(7 . 6 . 5 . 4 . 3 . 2 . 1)
=10 . 9 . 8
_{10 P 3} = 720
Question 39.
Fifty-two athletes arc competing in a bicycle race. In how many orders can the bicyclists finish first, second, and third? (Assume there are no ties.)
Answer:
Question 40.
An employee at a pet store needs to catch 5 tetras in an aquarium containing 27 tetras. In how many groupings can the employee capture 5 tetras?
Answer:
Given,
An employee at a pet store needs to catch 5 tetras in an aquarium containing 27 tetras.
_{27 C} _{5} = (frac<27!><(27-5) !>)
= (frac<27!><22!>) . (frac<1><5!>)
= 27. 26 . 25 . 24 . … 7 . 6 . 5 . 4 . 3 . 2 . 1/(22 . 21. … 7 . 6 . 5 . 4 . 3 . 2 . 1)(5 . 4 . 3 . 2 . 1)
= 27 . 26 . 5 . 23
_{27 C} _{5} = 80730
Thus the number of combinations of 27 tetras taken 5 at the time is 80730.
Question 41.
CRITICAL THINKING
Compare the quantities _{50}C_{9} and _{50}C_{41} without performing an calculations. Explain your reasoning.
Answer:
Question 42.
CRITICAL THINKING
Show that each identity is true for any whole numbers r and n, where 0 ≤ r ≤ n.
a. _{n}C_{n} = 1
Answer:
_{n}C_{n} = n!/n!(n – n)!
_{n}C_{n} = n!/n!0!
= 1/0!
We know that,
0! = 1
= 1/1 = 1
Thus _{n}C_{n} = 1
REASONING
Complete the table for each given value of r. Then write an inequality relating _{n}P_{r} and _{n}C_{r}. Explain your reasoning.
Answer:
Question 45.
PROBLEM SOLVING
You and your friend are in the studio audience on a television game show. From an audience of 300 people, 2 people are randomly selected as contestants. What is the probability that you and your friend are chosen?
Answer:
Question 46.
PROBLEM SOLVING
You work 5 evenings each week at a bookstore. Your supervisor assigns you 5 evenings at random from the 7 possibilities. What is the probability that your schedule does not include working on the weekend?
Answer:
_{n}C_{a} = n!/a!(n – a)!
_{7 C} _{5} = (frac<7!><(7-5) !>)
= (frac<7!><2!>) . (frac<1><5!>)
=7 . 6 . 5 . 4 . 3 . 2 . 1/(2 . 1)(5 . 4 . 3 . 2 . 1)
= 7 . 3
_{7 C} _{5} = 21
Thus we see that there are 21 possible combinations of five days formed from 7 days.
Hence we see that one of 21 possible combination of 5 days does not contain saturday and sunday.
P = P(Are chosen one combination of 21 possible)
= Number of favorable outcomes/Number of possible outcomes
= 1/21
REASONING
In Exercises 47 and 48, find the probability of winning a lottery using the given rules. Assume that lottery numbers are selected at random.
Question 47.
You must correctly select 6 numbers, each an integer from 0 to 49. The order is not important.
Answer:
Question 48.
You must correctly select 4 numbers, each an integer from 0 to 9. The order is important.
Answer:
_{n}C_{a} = n!/a!(n – a)!
_{10 C} _{4} = (frac<10!><(10-4) !>)
= (frac<10!><6!>) . (frac<1><4!>)
=10 . 9 . 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1/(6 . 5 . 4 . 3 . 2 . 1)(4 . 3 . 2 . 1)
= 10 . 3 . 7
_{10 C} _{4} = 210
There are 210 possible combination of 10 numbers taken 4 at a time.
P (You winning a lottery) = P(Are chosen one combination 210 possible)
= Number of favorable outcomes/Number of possible outcomes
= 1/210
Question 49.
MATHEMATICAL CONNECTIONS
A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. Consider a convex polygon with n sides.
a. Use the combinations formula to write an expression for the number of diagonals in an n-sided polygon.
b. Use your result from part (a) to write a formula for the number of diagonals of an n-sided convex polygon.
Answer:
Question 50.
PROBLEM SOLVING
You are ordering a burrito with 2 main ingredients and 3 toppings. The menu below shows the possible choices. How many different burritos are possible
Answer:
You are ordering a burrito with 2 main ingredients and 3 toppings.
Total number of main ingredients = 6
As the order in which the ingredients are chosen is not important, the total number of ways to select 2 main ingredients out of 6 can be found by using the combinations formula.
_{n}C_{a} = n!/a!(n – a)!
_{6 C} _{2} = (frac<6!><(6-2) !>)
= (frac<6!><4!>) . (frac<1><2!>)
=6 . 5 . 4 . 3 . 2 . 1/(4 . 3 . 2 . 1)(2 . 1)
= 3 . 5
_{6 C} _{2} = 15
Total number of toppings = 8
Number of toppings to be chosen = 3
_{n}C_{a} = n!/a!(n – a)!
_{8 C} _{3} = (frac<8!><(8-3) !>)
= (frac<8!><5!>) . (frac<1><3!>)
=8 . 7 . 6 . 5 . 4 . 3 . 2 . 1/(5 . 4 . 3 . 2 . 1)(3 . 2 . 1)
= 8 . 7
_{8 C} _{3} = 56
Total possible selections = ways to select main ingredients × Ways to select toppings
= 15 × 56
= 840
Question 51.
PROBLEM SOLVING
You want to purchase 2 different types of contemporary music CDs and 1 classical music CD from the music collection shown. How many different sets of music types can you choose for your purchase?
a. How many combinations of three marbles can be drawn from the bag? Explain.
b. How many permutations of three marbles can be drawn from the bag? Explain.
Answer:
Question 52.
HOW DO YOU SEE IT?
A bag contains one green marble, one red marble, and one blue marble. The diagram shows the possible outcomes of randomly drawing three marbles from the hag without replacement.
a. How many combinations of three marbles can be drawn from the bag? Explain.
Answer:
_{n}C_{a} = n!/a!(n – a)!
_{3 C} _{3} = (frac<3!><(3-3) !>)
= (frac<3!><0!>) . (frac<1><3!>)
=3 . 2 . 1/(1)(3 . 2 . 1)
= 1
_{3 C} _{3} = 1
b. How many permutations of three marbles can be drawn from the bag? Explain.
Answer:
_{n}P_{a} = n!/(n – a)!
_{3 P} _{3} = (frac<3!><(3-3) !>)
= (frac<3!><0!>)
=3 . 2 . 1/1
= 6
_{3 P} _{3} = 3
There are 6 possible combinations.
Question 53.
PROBLEM SOLVING
Every student in your history class is required to present a project in front of the class. Each day, 4 students make their presentations in an order chosen at random by the teacher. you make your presentation on the first day.
a. What is the probability that you are chosen to be the first or second presenter on the first day ?
b. What is the probability that you are chosen to be the second or third presenter on the first day? Compare your answer with that in part (a).
Answer:
Question 55.
PROBLEM SOLVING
You are one of 10 students performing in a school talent show. The order of the performances is determined at random. The first 5 performers go on stage before the intermission.
a. What is the probability that you are the last performer before the intermission and your rival performs immediately before you?
b. What is the probability that you are not the first performer?
Answer:
Question 56.
THOUGHT PROVOKING
How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3, and 4? Repetition of digits is allowed.
Answer:
Let fixed number 1 on the first place and in the second place can be one of 5 digits 0, 1, 2, 3 or 4. Because repetition of digits is allowed. In third and fourth place it can also be one of five digits.
Therefore we see that 5 . 5 . 5 = 125 integers which begin with 1, can be formed.
Second, If we fixed 2 on the first place, by the same logic as in the first part, we get that 5³ = 125 integers which begin with 2.
Next, if we fixed number 3 on the first place, we obtain 5³ = 125 integers which begin with 2.
5³ + 5³ + 5³ = 375
375 integers greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3, 4.
Question 57.
PROBLEM SOLVING
There are 30 students in your class. Your science teacher chooses 5 students at random to complete a group project. Find the probability that you and your 2 best friends in the science class are chosen to work in the group. Explain how you found your answer.
Answer:
Question 58.
PROBLEM SOLVING
Follow the steps below to explore a famous probability problem called the birthday problem. (Assume there are 365 equally likely birthdays possible.)
b. Generalize the results from part (a) by writing a formula for the probability P(n) that at least 2 people in a group of n people share the same birthday. (Hint: Use _{n}P_{r} notation in your formula.)
Answer:
Based on the explanation in the part under a) we can conclude that the probability that at least two people share the same birthday in a group of n people is
P = 1 – _{365}P_{n}/365 n
c. Enter the formula from part (b) into a graphing calculator. Use the table feature to make a table of values. For what group size does the probability that at least 2 people share the same birthday first exceed 50%?
Answer:
/>
Based on the above table we see that for a group of 23 people and more, the probability that at least tweople share the same birthday exceed 50%
Maintaining Mathematical Proficiency
Question 59.
A bag contains 12 white marbles and 3 black marbles. You pick 1 marble at random. What is the probability that you pick a black marble?
Answer:
Given,
A bag contains 12 white marbles and 3 black marbles. You pick 1 marble at random
Question 60.
The table shows the result of flipping two coins 12 times. For what outcome is the experimental probability the same as the theoretical probability?
Answer:
The table shows the result of flipping two coins 12 times
P(HH) = P(HT) = P(TH) = P(TT) = 1/2 . 1/2 = 1/4
P(HH) = Number of favorable outcomes/Number of possible outcomes
= 2/12 = 1/6
P(HT) = 6/12 = 1/2
P(TH) = 3/12 = 1/4
P(TT) = 1/12
The most likely fell first heads, and second tails. Also, with the least probability fell twice tails.
12.6 Binomial Distributions
Exploration 1
Work with a partner: The histograms show the results when n coins are flipped.
STUDY TIP
When 4 coins are flipped (n = 4), the possible outcomes are
TTTT TTTH TTHT TTHH
THTT THTH THHT THHH
HTTT HTTH HTHT HTHH
HHTT HHTH HHHT HHHH.
The histogram shows the numbers of outcomes having 0, 1, 2, 3, and 4 heads.
a. In how many ways can 3 heads occur when 5 coins are flipped?
Answer:
b. Draw a histogram that shows the numbers of heads that can occur when 6 coins are flipped.
Answer:
c. In how many ways can 3 heads occur when 6 coins are flipped?
Answer:
Exploration 2
Determining the Number of Occurrences
a. Complete the table showing the numbers of ways in which 2 heads can occur when n coins are flipped.
Answer:
b. Determine the pattern shown in the table. Use your result to find the number of ways in which 2 heads can occur when 8 coins are flipped.
LOOKING FOR A PATTERN
To be proficient in math, you need to look closely to discern a pattern or structure.
Answer:
Question 3.
How can you determine the frequency of each outcome of an event?
Answer:
Question 4.
How can you use a histogram to find the probability of an event?
Answer:
Lesson 12.6 Binomial Distributions
Monitoring Progress
An octahedral die has eight sides numbered 1 through 8. Let x be a random variable that represents the sum when two such dice are rolled.
Question 1.
Make a table and draw a histogram showing the probability distribution tor x.
Answer:
Question 2.
What is the most likely sum when rolling the two dice?
Answer:
Question 3.
What is the probability that the sum of the two dice is at most 3?
Answer:
According to a survey, about 85% of people ages 18 and older in the U.S. use the Internet or e-mail. You ask 4 randomly chosen people (ages 18 and older) whether they use the Internet or email.
Question 4.
Draw a histogram of the binomial distribution for your survey.
Answer:
Question 5.
What is the most likely outcome of your survey?
Answer:
Question 6.
What is the probability that at most 2 people you survey use the Internet or e-mail?
Answer:
Exercise 12.6 Binomial Distributions
Vocabulary and Core Concept Check
Question 1.
VOCABULARY
What is a random variable?
Answer:
A random variable is a variable whose value is determined by the outcomes of a probability experiment.
Question 2.
WRITING
Give an example of a binomial experiment and describe how it meets the conditions of a binomial experiment.
Answer:
We flipping a coin 10 times and register what fell.
We know that events, coin toss are independent.
Each trial has only two possible outcomes: H and T
The probabilities are P(H) = P(T) = 1/2 and are the same for each trial.
We can conclude that this experiment is a binomial experiment.
Monitoring Progress and Modeling with Mathematics
In Exercises 3 – 6. make a table and draw a histogram showing the probability distribution for the random variable.
Question 3.
x = the number on a table tennis ball randomly chosen from a bag that contains 5 balls labeled ,” 3 halls labeled “2,” and 2 balls labeled .”
Answer:
Question 4.
c = 1 when a randomly chosen card out of a standard deck of 52 playing cards is a heart and c = 2 otherwise.
Answer:
Let C be a random variable that represents the randomly chosen card.
Standard desk have 52 playing cards.
P(C = 1) = Number of favorable outcomes/Total number of outcomes
= A randomly chosen card is a hard/Total number of cards
= 13/52
On the other hand,
P(C = 2) = A randomly chosen card is not a hard/Total number of cards
= 39/52
Question 5.
w = 1 when a randomly chosen letter from the English alphabet is a vowel and w = 2 otherwise.
Answer:
Question 6.
n = the number of digits in a random integer from O through 999.
Answer:
There are 10 outcomes for value 1, 90 outcomes for value 2, and 900 values for value 3.
P(N = 1) = Number of favorable outcomes/Total number of outcomes
= A randomly chosen integers in a one-digit number/Total number of integers
= 10/1000
= 1/100
P(N = 2) = 90/1000 = 9/100
P(N = 3) = 900/1000 = 9/10
In Exercises 7 and 8, use the probability distribution to determine (a) the number that is most likely to be spun on a spinner, and (b) the probability of spinning an even number.
Question 7.
Answer:
Question 8.
Answer:
The most likely number to be spun on the spinner is the value of random variable X(Number of spinner) of which P(X) is greatest.
From given histogram we see that this probability is greatest for X = 5.
Hence the most likely number to be spun on the spinner is 5.
P(Spinning an even number) = P(X = 10) + P(X = 20) = 1/6 + 1/12 = 1/4
USING EQUATIONS
In Exercises 9 – 12, calculate the probability of flipping a coin 20 times and getting the given number of heads.
Question 9.
1
Answer:
Question 10.
4
Answer:
P(Four success) = _{20}C_{4}(1/2) 4 (1/2) 20-4
= 20!/4!(20 – 4)!(1/2) 20
= 20!/4!(16)!(1/2) 20
= 0.0046
We see that the obtained probability is small, which is logical.
Question 11.
18
Answer:
Question 12.
20
Answer:
P(Four success) = _{20}C_{20}(1/2) 20 (1/2) 20-20
= 20!/20!(20 – 20)!(1/2) 20
=(1/2) 20
= 0.00000095
Question 13.
MODELING WITH MATHEMATICS
According to a survey, 27% of high school students in the United States buy a class ring. You ask 6 randomly chosen high school students whether they own a class ring.
a. Draw a histogram of the binomial distribution for your survey.
b. What is the most likely outcome of your survey?
c. What is the probability that at most 2 people have a class ring?
Answer:
Question 14.
MODELING WITH MATHEMATICS
According to a survey, 48% of adults in the United States believe that Unidentified Flying Objects (UFOs) are observing our planet. You ask 8 randomly chosen adults whether they believe UFOs are watching Earth.
a. Draw a histogram of the binomial distribution for your survey.
Answer:
p = P(the American is a sports fan) = 48% = 0.48
1 – p = P(the American is not a sports fan) = 1 – 0.48= 0.52
P (0 success) = _{0}C_{8} p 0 (1 – p) 8-0
= 8!/0!(8 – 0)! 1 . 0.52 8
= 0.52 8
= 0.1513
P (One person believe that UFOs are watching Earth) =_{8}C_{1} p¹(1 – p) 8-1
=0.03948
P (Two person believe that UFOs are watching Earth) =_{8}C_{2} p 2 (1 – p) 8-2
=0.1275
P (Three person believe that UFOs are watching Earth) =_{8}C_{3} p 3 (1 – p) 8-3
=0.2355
P (Four person believe that UFOs are watching Earth) =_{8}C_{4} p 4 (1 – p) 8-4
=0.2717
P (Five person believe that UFOs are watching Earth) =_{8}C_{5} p 5 (1 – p) 8-5
=0.2006
P (Six person believe that UFOs are watching Earth) =_{8}C_{6} p 6 (1 – p) 8-6
=0.0926
P (Seven person believe that UFOs are watching Earth) =_{8}C_{7} p 7 (1 – p) 8-7
=0.0244
P (Eight person believe that UFOs are watching Earth) =_{8}C_{8} p 8 (1 – p) 8-8
=0.0028
b. What is the most likely outcome of your survey?
Answer:
P (Four person believe that UFOs are watching Earth) = 0.2717
This probability has the highest, so we conclude that the most likely outcome is that four of the eight adults believe that UFOs are watching Earth.
c. What is the probability that at most 3 people believe UFOs are watching Earth?
Answer:
P (At most 3 persons believe that UFOs are watching Earth) = P (One person believe that UFOs are watching Earth) + P (Two person believe that UFOs are watching Earth) + P (Three person believe that UFOs are watching Earth)
= 0.0053 + 0.0395 + 0.1275 + 0.2355
= 0.4078
ERROR ANALYSIS
In Exercises 15 and 16, describe and correct the error in calculating the probability of rolling a 1 exactly 3 times in 5 rolls of a six-sided die.
Question 15.
Answer:
Question 16.
Answer:
Question 17.
MATHEMATICAL CONNECTIONS
At most 7 gopher holes appear each week on the farm shown. Let x represent how many of the gopher holes appear in the carrot patch. Assume that a gopher hole has an equal chance of appearing at any point on the farm.
a. Find P(x) for x = 0, 1, 2 …. 7.
Answer:
p = P(the gopher holes appear in the carrot patch)
= Area marked for carrot/Area of the whole farm
= Area of a square + Area of a triangle/Area of the whole farm
= 0.28125
1 – p = P(The gopher holes do not appear in the carrot patch)
= 1 – 0.28125
= 0.71875
P (0 success) = _{0}C_{7} p 0 (1 – p) 7-0
= 7!/0!(7 – 0)! 1 . 0.72 7
= 0.72 7
= 0.099
P (There is one gopher hole in the carrot patch) =_{7}C_{1} p¹(1 – p) 7-1
=0.27143
P (There is two gopher hole in the carrot patch) =_{7}C_{2} p¹(1 – p) 7-2
=0.31863
P (There is three gopher hole in the carrot patch) =_{7}C_{3} p¹(1 – p) 7-3
=0.20781
P (There is four gopher hole in the carrot patch) =_{7}C_{4} p¹(1 – p) 7-4
=0.08131
P (There is five gopher hole in the carrot patch) =_{7}C_{5} p¹(1 – p) 7-5
=0.01909
P (There is six gopher hole in the carrot patch) =_{7}C_{6} p¹(1 – p) 7-6
=0.00249
P (There is seven gopher hole in the carrot patch) =_{7}C_{7} p¹(1 – p) 7-7
=0.00012
b. Make a table showing the probability distribution for x.
c. Make a histogram showing the probability distribution for x.
Answer:
Question 18.
HOW DO YOU SEE IT?
Complete the probability distribution for the random variable x. What is the probability the value of x is greater than 2?
Answer:
P(X = 1) + P(X = 2) + … + P(X = n) = 1
P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 0.1 + 0.3 + 0.4 + P(X = 4)
= 1
P(X = 4) = 1 – 0.8 = 0.2
Now lets find a probability that value of X is greater then two, as
P(X ≥ 2) = P(X = 2) + P(X = 3) + P(X = 4)
= 0.3 + 0.4 + 0.2
= 0.9
It is very likely that the random variable k will take a value greater than 2.
Question 19.
MAKING AN ARGUMENT
The binomial distribution Shows the results of a binomial experiment. Your friend claims that the probability p of a success must be greater than the probability 1 – p of a failure. Is your friend correct? Explain your reasoning.
Answer:
Question 20.
THOUGHT PROVOKING
There are 100 coins in a bag. Only one of them has a date of 2010. You choose a coin at random, check the date, and then put the coin back in the bag. You repeat this 100 times. Are you certain of choosing the 2010 coin at least once? Explain your reasoning.
Answer:
Given,
There are 100 coins in a bag. Only one of them has a date of 2010. You choose a coin at random, check the date, and then put the coin back in the bag. You repeat this 100 times
p = P(Selected coin has a date of 2010)
= Number of favorable outcomes/Total number of outcomes
= 1/100
1 – p = 1 – 1/100 = 99/100
P (0 success) = _{100}C_{0} p 0 (1 – p) 100-0
= 100!/0!(100 – 0)! 1 . (99/100) 100
= (99/100) 100
= 0.366
P(1 or more success) = 1 – P(0 success) = 1 – 0.366 = 0.634
Hence with a probability of 0.634, we will choose a coin which has a date of 2010.
We are not certain of choosing the 2010 coin at least once.
Question 21.
MODELING WITH MATHEMATICS
Assume that having a male and having a female child are independent events, and that the probability of each is 0.5.
a. A couple has 4 male children. Evaluate the validity of this statement: “The first 4 kids were all boys, so the next one will probably be a girl.”
b. What is the probability of having 4 male children and then a female child?
c. Let x be a random variable that represents the number of children a couple already has when they have their first female child. Draw a histogram of the distribution of P(x) for 0 ≤ x ≤ 10. Describe the shape of the histogram.
Answer:
Question 22.
CRITICAL THINKING
An entertainment system has n speakers. Each speaker will function properly with probability p. independent of whether the other speakers are functioning. The system will operate effectively when at least 50% of its speakers are functioning. For what values of p is a 5-speaker system more likely to operate than a 3-speaker system?
Answer:
Given,
An entertainment system has n speakers. Each speaker will function properly with probability p. independent of whether the other speakers are functioning.
The system will operate effectively when at least 50% of its speakers are functioning.
p = P(Speaker will function properly)
1 – 9 = P(Speaker will not function properly)
P(5-speaker system operate) = P(X = 3) + P(X = 4) + P(X = 5)
P(5-Speaker system operate) = P (0 success) = _{5}C_{3} p 3 (1 – p) 5-3 + _{5}C_{4} p 4 (1 – p) 5-4 + _{5}C_{5} p 5 (1 – p) 5-5 = 10p 3 (1 – p) 5-4 + 5p 4 (1 – p) 5
P = P(X = 2) + P(X = 3)
= _{5}C_{2} p 2 (1 – p) 5-2 + _{5}C_{3} p 3 (1 – p) 5-3
= 10p 2 (1 – p) 5-3 + 10p 3 (1 – p) 2
10p 3 (1 – p) 5-4 + 5p 4 (1 – p) 5 + p 5 > 10p 2 (1 – p) 3 + 10p 3 (1 – p) 2
5p 2 – 4p 3 – 10 + 30p – 30p 2 + 10p 3 > 0
A 5-speaker system operate more likely to operate than 3-speaker system when p ∈ (0.558, 1]
Maintaining Mathematical Proficiency
List the possible outcomes for the situation.
Question 23.
guessing the gender of three children
Answer:
Question 24.
picking one of two doors and one of three curtains
Answer:
If we denote by D1 and D2 first and second door and with C1, C2, C3 first, second and third curtain, then the possible outcomes are
D1C1, D1C2, D1C3, D2C1, D2C2, D2C3
Thus there are six possible outcomes.
Probability Review
12.1 Sample Spaces and Probability
Question 1.
A bag contains 9 tiles. one for each letter in the word HAPPINESS. You choose a tile at random. What is the probability that you choose a tile with the letter S? What is the probability that you choose a tile with a letter other than P?
Answer:
Let X be a random variable that represent the letter on tile.
We know that a bag contains tiles labeled with “H”, “A”, “P”, “I”, “N”, “E” and “S”
So we can conclude that the possible values for X are letters “H”, “A”, “P”, “I”, “N”, “E” and “S” and total number of outcomes is 9.
Question 2.
You throw a dart at the board shown. Your dart is equally likely to hit any point inside the square board. Are you most likely to get 5 points, 10 points, or 20 points?
Answer: It is the most likely to get 20 points.
Explanation:
From given board, we can conclude that the probability that we get 5 points is
P(5 points) = Surface of red area/Surface of board = 4/36 = 1/9
On the other hand, we see that the probability we get 10 points is
P(10 points) = Surface of yellow area/Surface of board = (16 – 4)/36 = 1/3
and the probability that we get 20 points is
P(20 points) = Surface of blue area/Surface of board = (36 – 16)/36 = 5/9
From the obtained results we get that it is the most likely to get 20 points, which is logical because the surface of the blue area is the largest.
12.2 Independent and Dependent Events
Find the probability of randomly selecting the given marbles from a bag of 5 red, 8 green, and 3 blue marbles when (a) you replace the first marble before drawing the second, and (b) you do not replace the first marble. Compare the probabilities.
Question 3.
red, then green
Answer:
We selected two marbles from a bag of 5 red, 8 green and 3 blue. With R denote the event that the red marble is drawn, with P blue, and with G green. The draws are independent because we replace the first marble before drawing the second. Therefore, the probability that we selected first red marble and then green is
P(RG) = P(R)P(G) = 5/16 . 8/16 = k5/36 = 0.15625
b. In this case, we do not replace the first marble before drawing the second. So, the draws are not independent.
P(RG) = P(R)P(G|R) = 5/16 . 8/15 = 1/6 = 0.16667
It is more likely that we selected first red marble and then green when we not replace the first marble before drawing the second.
Question 4.
blue, then red
Answer:
We selected two marbles from a bag of 5 red, 8 green and 3 blue. With R denote the event that the red marble is drawn, with P blue, and with G green. The draws are independent because we replace the first marble before drawing the second. Therefore, the probability that we selected first red marble and then green is
P(BR) = P(B)P(R) = 3/16 . 5/16 = 15/256 = 0.05859
b. In this case, we do not replace the first marble before drawing the second. So, the draws are not independent.
P(BR) = P(B)P(R|B) = 3/16 . 5/15 = 1/16 = 0.0625
It is more likely that we selected first blue marble and then red when we not replace the first marble before drawing the second.
Question 5.
green, then green
Answer:
We selected two marbles from a bag of 5 red, 8 green and 3 blue. With R denote the event that the red marble is drawn, with P blue, and with G green. The draws are independent because we replace the first marble before drawing the second. Therefore, the probability that we selected first red marble and then green is
P(GG) = P(G)P(G) = 8/16 . 8/16 = 1/4 = 0.25
b. In this case, we do not replace the first marble before drawing the second. So, the draws are not independent.
P(GG) = P(G)P(G|G) = 8/16 . 7/15 = 0.23333
It is more likely that we selected first blue marble and then red when we not replace the first marble before drawing the second.
12.3 Two-Way Tables and Probability
Question 6.
What is the probability that a randomly selected resident who does not support the project in the example above is from the west side?
Answer:
P(West side|Does not support the project) = P(West side and Does not support the project)/P(Does not support the project)
= 0.09/(0.08 + 0.09)
= 0.529
The probability that random selected resident who does not support the project is from the west side is about 0.529
Question 7.
After a conference, 220 men and 270 women respond to a survey. Of those, 200 men and 230 women say the conference was impactful. Organize these results in a two-way table. Then find and interpret the marginal frequencies.
Answer:
After a conference, 220 men and 270 women respond to a survey. Of those, 200 men and 230 women say the conference was impactful.
Number of men who say the conference had not impact = 220 – 200 = 20
By the same method we come to the conclusion
Number of women who say the conference had not impact = 270 – 230 = 40
Now, we will find the marginal frequencies.
Total number of people who say the conference was impact is 200 + 230 = 430
Total Number of people who say the conference had not impact = 20 + 40 = 60
Also from given information we know that total number of people who was surveyed is 220 + 270 = 490
12.4 Probability of Disjoint and Overlapping Events
Question 8.
Let A and B be events such that P(A) = 0.32, P(B) = 0.48, and P(A and B) = 0.12. Find P(A or B).
Answer:
P(A or B) = P(A) + P(B) – P(A and B)
Given,
P(A) = 0.32, P(B) = 0.48, and P(A and B) = 0.12
P(A or B) = 0.32 + 0.48 – 0.12 = 0.68
Question 9.
Out of 100 employees at a company, 92 employees either work part time or work 5 days each week. There are 14 employees who work part time and 80 employees who work 5 days each week. What is the probability that a randomly selected employee works both part time and 5 days each week?
Answer:
A =
B =
Based on the given information we see that
P(A) = Number of favorable outcomes/Total Number of outcomes
= Number of Employees either work part time/Total number of employees
= 14/100
= 0.14
P(B) = Number of favorable outcomes/Total Number of outcomes
= Number of Employees either work 5 days/Total number of employees
= 80/100
= 0.8
Also, we know that 92 employees either work part time or 5 days each week
P(A or B) = Number of favorable outcomes/Total Number of outcomes
= 92/100 = 0.92
For given events A and B the probability of A or B is
P(A or B) = P(A) + P(B) – P(A and B)
P(A or B) = 0.14 + 0.8 – 0.92 = 0.02
Hence the probability that a randomly selected employee works part time and 5 days each week is 0.02
12.5 Permutations and Combinations
Question 10.
_{7}P_{6}
Answer:
We know that number of permutations of n objects taken at a time (a ≤ n)
_{n}P_{a} = n!/(n – a)!
= 7!/(7 – 6)!
= 7 . 6 . 5 . 4 . 3 . 2 . 1
= 5040
_{7}P_{6} = 7! = 5040
Question 11.
_{13}P_{10}
Answer:
We know that number of permutations of n objects taken at a time (a ≤ n)
_{n}P_{a} = n!/(n – a)!
= 13!/(13 – 10)!
= 13!/3!
= 13. 12 . 11 . 10 . 9 . 8 . 7 . 6 . 5 . 4 = 94348800
_{13}P_{10} = 94348800
Question 14.
Eight sprinters are competing in a race. How many different ways can they finish the race? (Assume there are no ties.)
Answer:
_{n P} _{n} = n!
We know thata 8 sprinters are participating in a race. It is important to us in what order each of them will reach the goal. This tells us that it is necessary to calculate the number of permutations of 8 sprinters.
_{8 P} _{8} = 8! = 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1 = 40320
Eight sprinters can finish a race in 40320 ways.
Question 15.
A random drawing will determine which 3 people in a group of 9 will win concert tickets. What is the probability that you and your 2 friends will win the tickets?
Answer:
_{n}C_{a} = n!/a!(n – a)!
_{9}C_{3} = 9!/3!(9 – 3)!
= 9!/3!(6)!
= 9 . 8 . 7 6 . 5 . 4 . 3 . 2 . 1/(3 . 2 . 1)(6 . 5 . 4 . 3 . 2 . 1)
= 84
_{9}C_{3} = 84
The probability that you and your 2 friends will win the tickets is equal to the probability that out of 84 possibilities, the trio in which you and your friend are in will be chosen.
P = Number of favorable outcomes/ Total number of outcomes = 1/84
12.6 Binomial Distributions
Question 16.
Find the Probability of flipping a coin 12 times and getting exactly 4 heads.
Answer:
The probabilities are P(H) = P(T) = 1/2, that is p = p – 1 = 1/2 and are the same for each trial.
n = 12
P = 12!/4!(12 – 4)! . (1/2) 12
= 0.1208
Therefore the probability of flipping a coin 12 times and getting exactly 4 heads is about 0.12
Question 17.
A basketball player makes a free throw 82.6% of the time. The player attempts 5 free throws. Draw a histogram of the binomial distribution of the number of successful free throws. What is the most likely outcome?
Answer:
Given,
A basketball player makes a free throw 82.6% of the time. The player attempts 5 free throws.
p = P(Successful free throw) = 82.6% = 0.826
1 – p = P(Unsuccessful free throw) = 1 – 0.826 = 0.174
P (Out of 5 free throws one was successful) = _{5}C_{1} p¹(1 – p) 5-1
P (Out of 5 free throws two was successful) =_{5}C_{2} p²(1 – p) 5-2
P(Out of 5 free throws three was successful) = _{5}C_{3}p³(1 – p) 5-3
P (Out of 5 free throws four was successful) =_{5}C_{4} p 4(1 – p)5-4
P (All 5 throws one was successful) = _{5}C_{5}(1 – p)5-5
Probability Test
You roll a six-sided die. Find the probability of the event described. Explain your reasoning.
Question 1.
You roll a number less than 5.
Answer:
The die has 6 sides, thus the total number of possible outcomes is 6.
The favorable outcomes are
P(n<5) = Number of favorable outcomes/Total number of outcomes
Thus there are 4 favorable outcomes.
The probability to roll a number less than 5 is
= 4/6
= 2/3
Question 2.
You roll a multiple of 3.
Answer:
The die has 6 sides, thus the total number of possible outcomes is 6.
The favorable outcomes are
P(n = 3k) = Number of favorable outcomes/Total number of outcomes
Thus there are 2 favorable outcomes.
The probability to roll a number less than 3 is
= 2/6
= 1/3
Question 3.
_{7}P_{2}
Answer:
We know that number of permutations of n objects taken at a time (a ≤ n)
_{n}P_{a} = n!/(n – a)!
_{7}P_{2} = 7!/(7 – 2)!
= 7!/5!
= (7 . 6 . 5 . 4 . 3 . 2 . 1)/(5 . 4 . 3 . 2 . 1)
= 7 . 6
= 42
_{7}P_{2} = 42
Question 4.
_{8}P_{3}
Answer:
We know that number of permutations of n objects taken at a time (a ≤ n)
_{n}P_{a} = n!/(n – a)!
_{8}P_{3} = 8!/(8 – 3)!
= 8!/5!
= (8 . 7 . 6 . 5 . 4 . 3 . 2 . 1)/(5 . 4 . 3 . 2 . 1)
= 8 . 7 . 6
= 336
_{8}P_{3} = 336
Question 6.
_{12}C_{7}
Answer:
_{n}C_{a} = n!/a!(n – a)!
_{12}C_{7} = 12!/7!(12 – 7)!
= 12!/7!(5)!
= 12 . 11 . 10 . 9 . 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1/(7 . 6 . 5 . 4 . 3 . 2 . 1)(5 . 4 . 3 . 2 . 1)
= 11 . 2 . 9 . 4 = 792
_{12}C_{7} = 792
Question 7.
In the word PYRAMID, how many ways can you arrange
(a) all of the letters and
Answer:
We have to find the number of permutation all of the letters in a given word that will consist of 7 letters. We see that it matters that order of letters are important.
Number of permutations = (In the 1st place can be one of 7 letters) × (In the 2nd place can be one of 6 letters that is left)
(In the 3rd can be one of 5 letters that is left) × (In the 4th can be 4 letters that is left)
(In the 5th can be three letter that is left) × (In the 6th can be two letters that is left)
(In the 7th can be one letter that is left)
= 7 . 6 . 5 . 4 . 3 . 2 . 1 = 5040
Therefore, we have 5040 ways for arrange all of the letters in given word, that is PYRAMID, PYRAMDI, PYRADMI, ……., DIMARYP.
(b) 5 of the letters?
Answer:
We have to find the number of permutation all of the letters in a given word that will consist of 3 letters. We see that it matters that order of letters are important.
Number of permutations = (In the 1st place can be one of 7 letters) × (In the 2nd place can be one of 6 letters that is left)
(In the 3rd can be one of 5 letters that is left) × (In the 4th can be 4 letters that is left)
(In the 5th can be three letter that is left)
= 7 . 6 . 5 . 4 . 3
= 2520
Therefore, we have 2520 ways for arrange 5 of the letters in given word, that is PYRAM, PYRAI, PYRAD….
Question 8.
You find the probability P(A or B) by using hie equation P(A or B) = P(A) + P(B) – P(A and B). Describe why it is necessary to subtract P(A and B) when the events A and B are overlapping. Then describe why it is not necessary to subtract P(A and B) when the events A and B are disjoint.
Answer:
When the events are overlapping then P(A and B) ≠ 0. This means that the expression P(A) + P(B) already contains the probability P(A and B) and so P(A and B) is subtracted from their sum to evaluate P(A or B)
When the events are overlapping then P(A and B) = 0 and so the equation of P(A or B reduces to P(A) + P(B))
Question 9.
Is it possible to use the formula P(A and B) = P(A) • P(B/A) when events A and B are independent? Explain your reasoning.
Answer:
If events A and B are independent events, then
P(A and B) = P(A) . P(B)
Also, because event B is independent of event A then P(B|A) = P(A)
P(A and B) = P(A) . P(B) = P(A) . P(B|A)
where A and B are the independent events.
Question 10.
According to a survey, about 58% of families sit down tor a family dinner at least four times per week. You ask 5 randomly chosen families whether the have a family dinner at least four times per week.
a. Draw a histogram of the binomial distribution for the survey.
Answer:
p = P(Family have dinner four times per week) = 58% = 0.58
1 – p = P(Family have no dinner dour times per week) = 1 – 0.58 = 0.42
P (One Family have dinner four times per week) = _{5}C_{1} p¹(1 – p) 5-1
= 0.09024
P (Two Family have dinner four times per week) =_{5}C_{2} p²(1 – p) 5-2
= 0.24923
P(Three Family have dinner four times per week) = _{5}C_{3}p³(1 – p) 5-3
= 0.34418
P (Four Family have dinner four times per week) =_{5}C_{4} p 4(1 – p)5-4
= 0.23765
P (Family have dinner four times per week) = _{5}C_{5}(1 – p)5-5 = 0.06563
b. What is the most likely outcome of the survey?
Answer:
P(Three families have dinner four times per week) = 0.34418
This probability is the highest, so we can conclude that the most likely outcome is that three of the five families have dinner four times per week.
c. What is the probability that at least 3 families have a family dinner four times per week?
Answer:
In this part we have to find the probability that,
P(At least 3 families have dinner four times per week)
= P(Three families have dinner four times per week) + P(Four families have dinner four times per week) + P(Five families have dinner four times per week)
= 0.34418 + 0.23765 + 0.06563
= 0.64746
Question 11.
You are choosing a cell phone company to sign with for the next 2 years. The three plans you consider are equally priced. You ask several of your neighbors whether they are satisfied with their current cell phone company. The table shows the results. According to this survey, which company should you choose?
Answer:
To find the joint relative frequencies we divide each frequency by the total number of people in the survey. Also the marginal relative frequencies we find as the sum of each row and each column.
So, we can present a two way table that shows the joint and marginal relative frequencies.
Finally, to get conditional relative frequencies we use the previous the marginal relative frequency of each row.
Therefore we should choose company A.
Question 12.
The surface area of Earth is about 196.9 million square miles. The land area is about 57.5 million square miles, and the rest is water. What is the probability that a meteorite that reaches the surface of Earth will hit land? What is the probability that it will hit water?
Answer:
From the formula for geometric probability we get the probability that a meteorite that reaches the surface of Earth will hit lend is
P = The lend area/ The surface area of Earth = 57.5/196.9 = 0.292
Alos we know that the probability for complement of event A is
P((ar)) = 1 – P(A)
In this case, complement of event is .
P = 1 – 0.292 = 0.708
Question 13.
Consider a bag that contains all the chess pieces in a set, as shown in the diagram.
a. You choose one piece at random. Find the probability that you choose a black piece or a queen.
Answer:
The total number of pieces is 2 + 2 + 4 + 4 + 4 + 16 = 32.
Also we see that the number of black and white pieces are the same, 16 and there are one black queen.
A =
B =
P(A) = Number of favorable outcomes/Total number of outcomes = Number of black pieces/Total number of pieces = 16/32 = 1/2
P(B) = Number of queens/Total number of pieces = 2/32 = 1/16
P(A and B) = Number of black queens/Total number of pieces = 1/32
P(A or B) = P(A) + P(B) – P(A and B)
= 1/2 + 1/16 – 1/32 = 17/32
b. You choose one piece at random, do not replace it, then choose a second piece at random. Find the probability that you choose a king, then a pawn.
Answer:
C =
P(C) = Number of pawns/Total number of pieces = 16/32 = 1/2
P(C and D) = Number of pawns and king/Total number of pieces = (16 + 2)/32 = 9/16
We know that for two dependent events A and B probability that both occur is
P(A and B) = P(A)P(B|A)
P(D|C) = P(C and D)/P(C) = 1/2 × 16/9 = 8/9
Question 14.
Three volunteers are chosen at random from a group of 12 to help at a summer camp.
a. What is the probability that you, your brother, and your friend are chosen?
Answer:
_{n}C_{a} = n!/a!(n – a)!
_{12}C_{3} = 12!/3!(12 – 3)!
= 12!/3!(9)!
= 12 . 11 . 10 . 9 . 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1/(3 . 2 . 1)(9 . 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1)
= 12 . 11 . 10/3 . 2 . 1
= 2 . 11 . 10
_{12}C_{3} = 220
P = Number of favorable outcomes/Total number of outcomes = 1/220
b. The first person chosen will be a counselor, the second will be a lifeguard, and the third will be a cook. What is the probability that you are the cook, your brother is the lifeguard, and your friend is the counselor?
Answer:
We know that number of permutations of n objects taken at a time (a ≤ n)
_{n}P_{a} = n!/(n – a)!
_{12}P_{3} = 12!/(12 – 3)!
= 12!/9!
= (12 . 11 . 10 . 9 . 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1)/(9 . 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1)
= 12 . 11 . 10
= 1320
_{12}P_{3} = 1320
P = 1/1320
Probability Cumulative Assessment
Question 1.
According to a survey, 63% of Americans consider themselves sports fans. You randomly select 14 Americans to survey.
a. Draw a histogram of the binomial distribution of your survey.
Answer:
p = P(the American is a sports fan) = 63% = 0.63
1 – p = P(the American is not a sports fan) = 1 – 0.63= 0.37
P (0 success) = _{12}C_{0} p 0 (1 – p) 12-0
= 12!/0!(12 – 0)! 1 . 0.37 12
= 0.37 12
P (One American is a sports fans) =_{12}C_{1} p¹(1 – p) 12-1
=0.000134506
P (Two Americans are sports fans) =_{12}C_{2} p 2 (1 – p) 12-2
=0.001259628
P (Three Americans are sports fans) =_{12}C_{3} p 3 (1 – p) 12-3
=0.007149239
P (Four American is a sports fans) =_{12}C_{4} p 4 (1 – p) 12-4
=0.027389316
P (Five American is a sports fans) =_{12}C_{5} p 5 (1 – p) 12-5
=0.07461738
P (Six American is a sports fans) =_{12}C_{6} p 6 (1 – p) 12-6
=0.148226418
P (Seven American is a sports fans) =_{12}C_{7} p 7 (1 – p) 12-7
=0.216330447
P (Eight American is a sports fans) =_{12}C_{8} p 8 (1 – p) 12-8
=0.230216523
P (Nine American is a sports fans) =_{12}C_{9} p 9 (1 – p) 12-9
=0.174217909
P (Ten American is a sports fans) =_{12}C_{10}p 10 (1 – p) 12-10
=0.088992392
P (Eleven American is a sports fans) =_{12}C_{11} p 11 (1 – p) 12-11
=0.023755047
P (Twelve American is a sports fans) =_{12}C_{1} p 12 (1 – p) 12-12
=0.003909188
b. What is the most likely number of Americans who consider themselves Sports fans?
Answer:
From histogram in part a and based on the calculations in the first part
We know that,
P(Eight Americans are sports fans) = 0.23
This probability is highest, so we can conclude that the most likley outcome is that eight of the 12 selected Americans consider themselves sports fans.
c. What is the probability at least 7 Americans consider themselves sports fans?
Answer:
In this part we have to find dthe probability that
P(At least 7 Americans consider themselves sports fans)
= P(Seven American is a sports fans) + P(Eight American is a sports fans) + P(Nine American is a sports fans) + P(Ten American is a sports fans) + P(Eleven American is a sports fans) + P(Twelve American is a sports fans)
= 0.2163 + 0.2302 + 0.1742 + 0.08899 + 0.0275 + 0.0039
= 0.7412
Therefore, we can conclude that the probability that at least 7 Americans consider themselves sports fans is 0.7412
Question 2.
What is the arc length of (widehat) ?
(A) 3.5 π cm
(B) 7 π cm
(C) 21 π cm
(D) 42 π cm
Answer:
Arc length of (widehat) = 105/360 × 2π × 12 = 7π cm
Question 3.
you order a fruit smoothie made with 2 liquid ingredients and 3 fruit ingredients from the menu shown. How many different fruit smoothies can you order?
Answer:
Question 4.
The point (4, 3) is on a circle with center (- 2, – 5), What is the standard equation of the circle?
Answer:
Given,
The point (4, 3) is on a circle with center (- 2, – 5)
(x – h)² + (y – k)² = r²
Substitute the values in the given equation
(4 – (-2))² + (3 – (-5))² = r²
100 = r²
r = √100 = 10
General equation of circle
(x – h)² + (y – k)² = r²
Substitute the values in the given equation
(x – (-2))² + (y – (-5))² = 10²
(x + 2)² + (y + 5)² = 100
Question 5.
Find the length of each line segment with the given endpoints. Then order the line segments from shortest to longest.
a. A(1, – 5), B(4, 0)
Answer:
The general equation of distance between 2 points is
d = √(x2 – x1)² + (y2 – y1)²
dAB = √(4 – 1)² + (0 – (-5))²
= √34
= 5.83
b. C(- 4, 2), D(1, 4)
Answer:
The general equation of distance between 2 points is
d = √(x2 – x1)² + (y2 – y1)²
dCD = √(1 – (-4))² + (4 – 2)²
= √29
= 5.39
c. E(- 1, 1), F(- 2, 7)
Answer:
The general equation of distance between 2 points is
d = √(x2 – x1)² + (y2 – y1)²
dEF = √(-2 – (-1))² + (7 – 1)²
= √37
= 6.083
d. G(- 1.5, 0), H(4.5, 0)
Answer:
The general equation of distance between 2 points is
d = √(x2 – x1)² + (y2 – y1)²
dGH = √(4.5 – (-1.5))² + (0 – 0)²
= √36
= 6
e. J(- 7, – 8), K(- 3, – 5)
Answer:
The general equation of distance between 2 points is
d = √(x2 – x1)² + (y2 – y1)²
dJK = √(-3 – (-7))² + (-5 – (-8))²
= √25
= 5
f. L(10, – 2), M(9, 6)
Answer:
The general equation of distance between 2 points is
d = √(x2 – x1)² + (y2 – y1)²
dLM = √(9 – 10)² + (6 – (-2))²
= √65
= 8.06
Therefore, the line segments in ascending order are JK < CD < AB < GH < EF < LM
Question 6.
Use the diagram to explain why the equation is true.
P(A) + P(B) = P(A or B) + P(A and B)
Answer:
Given,
P(A) = 8/12
P(B) = 7/12
P(A or B) = 12/12
P(A and B) = 3/12
P(A) + P(B) = P(A or B) + P(A and B)
8/12 + 7/12 = 12/12 + 3/12
5/4 = 5/4
Question 7.
A plane intersects a cylinder. Which of the following cross sections cannot be formed by this intersection?
(A) line
(B) triangle
(C) rectangle
(D) circle
Answer:
If the plane is tangent to the curved surface of the cylinder, then the intersection is a line.
If the plane is cuts the cylinder parallel to its circular base, then the intersection is a circle.
If the plane cuts the cylinder perpendicular to its circular base and parallel to its curved surface, then the intersection is a rectangle.
The correct answer is option B.
Question 8.
A survey asked male and female students about whether the prefer to take gym class or choir. The table shows the results of the survey,
a. Complete the two-way table.
Answer:
From the given table we first find the joint frequencies.
Number of people who prefer gym – 49 = 57
On the other hand, we know that 57 people prefer gym class, of which 23 was female.
Number of male who prefer the gym – 23 = 34
Number of male who prefer the choir – 34 = 16
Number of female who prefer the choir – 16 = 33
Now, we will find the remaining marginal frequencies. Hence we know that 23 female prefer gym class and 33 female prefer choir class.
Total number of female is 23 + 33 = 56
b. What is the probability that a randomly selected student is female and prefers choir?
Answer:
P = Number of female who prefer choir class/Total number of students
= 33/106
= 0.31
c. What is the probability that a randomly selected male student prefers gym class?
Answer:
P(Prefer gym class|Male) = P(Male and prefer gym class)/P(Male)
= Number of male who prefer gym class/Number of male students
= 34/50
= 0.68
c. Suppose the least-reliable mower stops working completely. How does this affect the probability that the lawn-moving business can be productive on a given day?
Answer:
If the least-reliable mower stops working completely, this will not affect the probability that the lawn mowing business can be productive on a given day, because we know that as long as one of the mowers is working, the owner can stay productive.
Question 10.
You throw a dart at the board shown. Your dart is equally likely to hit any point inside the square board. What is the probability your dart lands in the yellow region?
(A) (frac
(B) (frac
(C) (frac
(D) (frac
Answer:
The total area of the given figure is (6(2))² = 144 sq. units
Area of the red circle = π × 2² = 4π
The area of the yellow ring is π × (4² – 2²) = 12π
The area of the blue ring is π × (6² – 4²) = 20π
Therefore, the probability of hitting the blue region is 12π/144 = π/12
Thus the correct answer is option B.
Hope you are all satisfied with the given solutions. You can get free access to Download Big Ideas Math Geometry Answers Chapter 12 Probability pdf from here. Bookmark our Big Ideas Math Answers to get detailed solutions for all Geometry Chapters.
6.1: Sample Spaces and Probability - Mathematics
The course will introduce the basic notion of probability theory and its application to statistics. The focus will be on the discussion of applications.
The text that will be used is:
Jay L. Devore, Probability and Statistics, Thomson
The syllabus can be found here.
There will be three midterms the first of which will be on Wednesday February 4th.
The exercises listed are only suggested. The more you do the better it is. In case of differences between the two books I will indicate in square brackets the number relative to the 5th edition.
Material covered: section 2.1 (Sample Spaces and Events), 2.2 (Axioms, Interpretations, and Properties of Probabilities), and 2.3 (Counting Techniques).
Suggested exercises: (2.1) 3,5,9, (2.2) 13, 17, 21 (2.3) 32, 35, 40
Material covered: section 2.4 (Conditional Probability), 2.5 (Independence), 3.1 (Random Variable)
Suggested exercises: (2.4) 50, 58, 63 (2.5) 78, 82, 85 [83] (Supplementary Exercises) 97, 104, 111 (3.1) 2, 7, 9.
Material covered: section 3.2 (Probability Distribution for a Discrete Random Variable), 3.3 (Expected Values of a Discrete Random Variable)
Suggested exercises: (3.2) 13, 22, 25 (3.3) 28, 29
Material covered: 3.3 (Expected Values of a Discrete Random Variable, cont.), 3.4 (The Binomial Probability Distribution), 3.5 (Hypergeometric and Negative Binomial Distributions),
Suggested exercises: (3.2) 37, 40, 43, (3.4) 52, 54, 62, (3.5) 66, 67 (3.6)
Material covered: 3.6 (The Poisson Probability Distribution)
Suggested exercises: (3.6) 83, 85
Material covered: 4.1 (Continuous Random Variable and Probability Distributions) 4.2 (Cumulative Distribution Functions and Expected Values) 4.3 (The Normal Distribution)
Suggested exercises: (4.1) 3, 5, 9 (4.2) 11, 15, 21 (4.3) 29, 31, 39, 47 [45]
Material covered: 4.4 (The Gamma Distribution and Its Relatives) 5.1 Jointly Distributed Random Variables) 5.2 (Expected Values, Covariance and Correlation)
Suggested exercises: (4.3) 33, 41, 45, 49, 53 (4.4) 59, 61, 63 (5.1) 1, 9, 15 (5.2) 23, 27, 35
Material covered: 1.1 (Population, Sample and Processes) 1.2 Pictorial and Tabular methods in Descriptive Statistics) 1.3 (Measure of Location) 1.4 (Measure of Variability) 5.3 (Statistics and Their Dstribution) 5.4 (Distribution of the Sample Mean)
Suggested exercises: (1.1) 3, 7 (1.2) 13, 17, 25, 27 (1.3) 35, 39, 41, (1.4) 45, 49, 53, 59, (5.3) 37, 41, 43 (5.4) 49, 51, 53, 56
Material covered: 6.1 (Some General Concepts of Point Estimation) 6.2 (Methods of Point Estimation [the subsection "The Method of Moments" was not covered])
Suggested exercises: (6.1) 1, 5, 7, 11, 17 (6.2) 25, 26, 29, 33, 37
Material covered: 7.1 (Basic Properties of Confidence Intervals) 7.2 (Large Sample Confidence Intervals for a Population Mean and Proportion)
Suggested exercises: (7.1) 3, 5, 9, 11 (7.2) 13, 17, 23, 25
February 4: Midterm. Practice test and solution. The test will cover up to the Hypergeometric Distribution. A review class where I'll solve the practice test is tentatively scheduled for Monday 3 form 5:00pm to 6:00pm pending availability of a classroom. The practice test contains the kind of question you may find in the midterm but is not meant as a guideline for the midterm.
The second midterm will be on March 3. Note the change of date due to the high number of people that will be out of town on March 5.
The test will cover sec. 3.6, 4.1, 4.2, 4.3, 5.1, 5.2. The property of linear functions and sums of Normal variable will be used.
The third midterm will be on April 14.
The makeup final is fixed for Tuesday April 27 at 11. We meet in my office. Only people with a good reason.
Here are few practice exercises for the final with solutions. I'll add the solutions and more exercises tomorrow. I I'll keep upgrading this page so I suggest you to check it.
Binomial Probability
In this section, we will consider types of problems that involve a sequence of trials, where each trial has only two outcomes, a success or a failure. These trials are independent. That is, the outcome of one does not affect the outcome of any other trial. Furthermore, the probability of success, p, and the probability of failure, (1 − p), remains the same throughout the experiment. These problems are called binomial probability problems. Since these problems were researched by a Swiss mathematician named Jacques Bernoulli around 1700, they are also referred to as Bernoulli trials.
We give the following definition:
Binomial Experiment: A binomial experiment satisfies the following four conditions:
- There are only two outcomes, a success or a failure, for each trial.
- The same experiment is repeated several times.
- The trials are independent that is, the outcome of a particular trial does not affect the outcome of any other trial.
- The probability of success remains the same for every trial.
The probability model that we are about to investigate will give us the tools to solve many real-life problems like the ones given below.
- If a coin is flipped 10 times, what is the probability that it will fall heads 3 times?
- If a basketball player makes 3 out of every 4 free throws, what is the probability that he will make 7 out of 10 free throws in a game?
- If a medicine cures 80% of the people who take it, what is the probability that among the 10 people who take the medicine, 6 will be cured?
- If a microchip manufacturer claims that only 4% of their chips are defective, what is the probability that among the 60 chips chosen, exactly three are defective?
- If a telemarketing executive has determined that 15% of the people contacted will purchase the product, what is the probability that among the 12 people who are contacted, 2 will buy the product?
We now consider the following example to develop a formula for finding the probability of k successes in n Bernoulli trials.
That's not fair!
In this unit we play probability games and learn about sample space and a sense of fairness.
- Use dice etc to assign roles and discuss the fairness of games.
- Play probability games and identify all possible outcomes.
- Compare and order the likelihood of simple events.
Three important ideas underpin this unit:
- The set of all possible outcomes of a random phenomenon is called the sample space.
- An event is any outcome, or set of outcomes of a random phenomenon.
- A fair game is a game in which there is an equal chance of winning or losing.
Students should be given lots of experience with spinners, coins, dice and other equipment that generates outcomes at random, such as drawing a name from a hat. The equipment can be used to play games, which should lead to a discussion of fairness (or otherwise) of the equipment and to finding the possible outcomes of using it. As they play games, record results and use the results to make predictions they find out that with probability they can never know exactly what will happen next, but they get an idea about what to expect.
Students at this Level will begin to explore the concept of equally likely events, such as getting a head or tail from the toss of a coin, or the spin of a spinner with two equal sized regions. Students can handle simple fractions at Level 2, and assigning simple probabilities provides them with an interesting and useful application of these numbers. Students can understand that the probability of getting a head when tossing a coin is 1/2. Given a spinner that is marked off equally in three colours, students can also understand that the probability of getting any one of the colours is 1/3 because there are three equally likely events and one of them has to happen.
This unit can be differentiated by varying the scaffolding of the tasks or altering expectations to make the learning opportunities accessible to a range of learners. For example:
- Work directly with students as they work through the probability games. Guide them to think through all possible outcomes, predict outcomes, record outcomes and reflect on results.
- Encourage students to work at their own pace taking as long as they need to work through each game. Students do not need to complete all of the games listed.
- Expect students to share their thinking about the fairness of the games, accepting that some students may be describing their experiences of playing the game rather than considering probability more generally.
Some of the activities in this unit can be adapted to use contexts and materials that are familiar and engaging for students. In particular, when students create their own games in the final session encourage them to consider their friends and classmates when planning, and to create a game that will appeal to them and be fun to play. This could be achieved by incorporating favourite elements from other games, or items of current interest.
- A large die (you can make one by cutting a large cube of foam rubber)
- Dice (one per student labelled 1 - 6)
- Copymasters of probability games
- Centimetre cubes
- Coins
We introduce the unit by rolling dice and investigating the numbers that come up.