### Chapter Review

## The Multiplication Rule for Counting

1
.

You are booking a round trip flight for vacation. If there are 4 outbound flight options and 7 return flight options, how many different options do you have?

2
.

You are putting together a social committee for your club. You’d like broad representation, so you will choose one person from each class. If there are 8 seniors, 12 juniors, 10 sophomores, and 6 first-years, how many committees are possible?

3
.

The Big Breakfast Platter at Jimbo’s Sausage Haus gives you your choice of 4 flavors of sausage, 5 preparations for eggs, 3 different potato options, and 4 different breads. If you choose one of each, how many different Big Breakfast Platters can be selected?

4
.

The multiple-choice quiz you’re about to take has 10 questions with 4 choices for each. How many ways are there to fill out the quiz?

## Permutations

5
.

Compute $\frac{8!}{2!3!3!}$.

6
.

Compute $\frac{12!}{8!3!}$.

7
.

Compute $\frac{211!}{210!}$

8
.

Compute ${}_{5}{P}_{3}$.

9
.

Compute ${}_{15}{P}_{3}$.

10
.

Compute ${}_{22}{P}_{5}$.

11
.

As you plan your day, you see that you have 6 tasks on your to-do list. You’ll only have time for 5 of those. How many schedules are possible for you today?

12
.

As captain of your intramural softball team, you are responsible for setting the 10-person batting order for the team. If there are 12 people on the team, how many batting orders are possible?

## Combinations

13
.

If you’re trying to decide which 4 of your 12 friends to invite to your apartment for a dinner party, are you using permutations or combinations?

14
.

If you’re trying to decide which of your guests sits where at your table, are you using permutations or combinations?

15
.

Compute ${}_{7}{C}_{4}$.

16
.

Compute ${}_{13}{C}_{8}$.

17
.

How many ways are there to draw a hand of 8 cards from a deck of 16 cards?

18
.

In a card game with 4 players and a deck of 12 cards, how many ways are there to deal out the four 3-card hands?

## Tree Diagrams, Tables, and Outcomes

19
.

If you draw a card at random from a standard 52-card deck and note its suit, what is the sample space?

20
.

If you draw 2 cards at random from a standard 52-card deck and note the 2 suits (without paying attention to the order), what is the sample space?

21
.

If you draw 2

*Scrabble*tiles in order without replacement from a bag containing E, E, L, S, what is the sample space?22
.

If you draw 2

*Scrabble*tiles without replacement and ignoring order from a bag containing E, E, L, S, what is the sample space?23
.
If you draw 2

*Scrabble*tiles without replacement and ignoring order from a bag containing E, E, L, S, what is the sample space?

24
.

If you draw 2

*Scrabble*tiles with replacement and ignoring order from a bag containing E, E, L, S, what is the sample space?## Basic Concepts of Probability

25
.

If you read that the probability of flipping 10 heads in a row is $\frac{1}{1024}$, is that probability most likely theoretical, empirical, or subjective?

26
.

If someone tells you that there is a 40% chance that a Democrat wins the U.S. Presidential election in 2132, is that probability most likely theoretical, empirical, or subjective?

27
.

If your professor says that you have a 20% chance of getting an A in her class because 20% of her students historically have earned As, is that probability most likely theoretical, empirical, or subjective?

In the following exercises, you are about to roll a standard 12-sided die (with faces labeled 1–12).

28
.

What is the probability of rolling a negative number?

29
.

What is the probability of rolling a number less than 20?

30
.

What is the probability of rolling an 11?

31
.

What is the probability of rolling a number less than 7?

32
.

What is the probability of

*not*rolling an 11?33
.

What is the probability of rolling a multiple of 4?

34
.

Over the last 30 years, it has rained 12 times on May 1. What empirical probability would you assign to the event "it rains next May 1"?

## Probability with Permutations and Combinations

In the following exercises, you’re drawing cards from a special deck of cards containing $2\u2661$, $2\mathrm{\u2663}$, $2\mathrm{\u2662}$, $2\mathrm{\u2660}$, $3\u2661$, $3\mathrm{\u2663}$, $3\mathrm{\u2660}$, $4\u2661$, $4\u2663$, $5\u2661$.

35
.

If you draw 4 cards without replacement, what is the probability of drawing a 2, 3, 4, and 5 in order?

36
.

If you draw 4 cards without replacement, what is the probability of drawing a 2, 3, 4, and 5 in

*any*order?37
.

If you draw 3 cards without replacement, what is the probability that you draw a $\u2661$, a $\u2663$, and a $\u2661$, in order?

38
.

If you draw 3 cards without replacement, what is the probability that you draw 2 $\u2661$ and 1 $\u2663$, in any order?

## What Are the Odds?

39
.

If you roll a standard 20-sided die (with faces numbered 1–20), what are the odds against rolling a number less than 5?

40
.

If you roll a standard 20-sided die (with faces numbered 1–20), what are the odds in favor of rolling greater than a 5?

41
.

If $P(E)=\frac{4}{7}$, what are the odds in favor of $E$?

42
.

If $P(E)=\frac{5}{17}$, what are the odds against $E$?

## The Addition Rule for Probability

In the following exercises, you’re drawing a single card from a special deck of cards containing $2\u2661$, $2\u2663$, $2\mathrm{\u2662}$, $2\u2660$, $3\u2661$, $3\u2663$, $3\u2660$, $4\u2661$, $4\u2663$, $5\u2661$.

43
.

What is the probability of drawing a 2 or a 3?

44
.

What is the probability of drawing a $\u2663$ or a $\u2660$?

45
.

What is the probability of drawing a 2 or a $\u2661$?

46
.

What is the probability of drawing an even number or a $\u2663$?

## Conditional Probability and the Multiplication Rule

In the following exercises, you’re drawing from a special deck of cards containing $2\u2661$, $2\u2663$, $2\mathrm{\u2662}$, $2\u2660$, $3\u2661$, $3\u2663$, $3\u2660$, $4\u2661$, $4\u2663$, $5\u2661$.

47
.

If you draw a single card, what is:

- $P(\text{draw a 2})$
- $P(\text{draw a 2}|\text{draw a}\mathrm{\u2661})$
- $P(\text{draw a 2}|\text{draw a}\mathrm{\u2660})$

48
.

If you draw a single card, what is:

- $P(\text{draw a}\mathrm{\u2661})$
- $P(\text{draw a}\mathrm{\u2661}|\text{draw a 3})$
- $P(\text{draw a}\mathrm{\u2661}|\text{draw a 2})$

In the following exercises, you are playing the following game that involves rolling 2 dice, one at a time. First, you roll a standard 6-sided die. If the result is a 4 or less, your second roll uses a standard 4-sided die. If the result of the first roll is a 5 or 6, your second roll uses a standard 6-sided die. Find these probabilities:

49
.

$P(\text{first roll is a 3 and second roll is a 3})$

50
.

$P(\text{first roll is a 6 and second roll is a 6})$

51
.

$P(\text{second roll is a 6})$

52
.

$P(\text{second roll is a 1})$

## The Binomial Distribution

In the following exercises, decide whether the described experiment is a binomial experiment. If it is, identify the number of trials and the probability of success in each trial. If it isn’t, explain why it isn’t.

53
.

Draw 5 cards with replacement from a standard deck and count the number of $\u2660$.

54
.

Draw 5 cards without replacement from a standard deck and count the number of $\u2660$.

55
.

Draw cards from a standard deck and count how many cards are chosen before the first $\u2660$ appears.

In the following exercises, you are about to roll a standard 20-sided die. Round answers to 4 decimal places.

56
.

Suppose you are going to roll the die 4 times. Give a full PDF table for the number of times a number greater than 16 appears.

57
.

If you roll the die 10 times, what is the probability that a number between 1 and 5 (inclusive) comes up exactly once?

58
.

If you roll the die 40 times, what is the probability that 20 comes up fewer than 2 times?

59
.

If you roll the die 40 times, what is the probability that 20 comes up 4 or more times?

60
.

If you roll the die 100 times, what is the probability that the number of times the die lands on something less than or equal to 7 is between 30 and 35 (inclusive)?

61
.

If you roll the die 100 times, what is the probability that the number of times the die lands on something less than or equal to 7 is exactly 36?

62
.

If you roll the die 100 times, what is the probability that the die lands on 20 between 5 and 8 times, inclusive?

## Expected Value

63
.

You are playing a game where you roll a pair of standard 6-sided dice. You win $32 if you get a sum of 12, and lose $1 otherwise. What is the expected value of this game?

64
.

Interpret your answer.

You are playing a game where you roll a standard 12-sided die 4 times. If you roll 12 four times, you win $1,000. If you roll 12 three times, you win $100. If you roll 12 twice, you win $10. If you roll 12 one time, you don’t win or lose anything. If you roll don’t roll a single 12, you lose $1.

65
.

What is the expected value of this game?

66
.

Interpret your answer.

67
.

Which game would be better to play? Why?