Contemporary Mathematics

# Chapter Review

### 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 $2♡$, $2\clubsuit$, $2\diamondsuit$, $2\spadesuit$, $3♡$, $3\clubsuit$, $3\spadesuit$, $4♡$, $4♣$, $5♡$.
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 $♡$, a $♣$, and a $♡$, in order?
38 .
If you draw 3 cards without replacement, what is the probability that you draw 2 $♡$ and 1 $♣$, 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 $2♡$, $2♣$, $2\diamondsuit$, $2♠$, $3♡$, $3♣$, $3♠$, $4♡$, $4♣$, $5♡$.
43 .
What is the probability of drawing a 2 or a 3?
44 .
What is the probability of drawing a $♣$ or a $♠$?
45 .
What is the probability of drawing a 2 or a $♡$?
46 .
What is the probability of drawing an even number or a $♣$?
##### Conditional Probability and the Multiplication Rule
In the following exercises, you’re drawing from a special deck of cards containing $2♡$, $2♣$, $2\diamondsuit$, $2♠$, $3♡$, $3♣$, $3♠$, $4♡$, $4♣$, $5♡$.
47 .

If you draw a single card, what is:

1. $P({\text{draw a 2}})$
2. $P({\text{draw a 2}}|{\text{draw a }}\heartsuit)$
3. $P({\text{draw a 2}}|{\text{draw a }}\spadesuit )$
48 .

If you draw a single card, what is:

1. $P({\text{draw a }}\heartsuit)$
2. $P({\text{draw a }}\heartsuit|{\text{draw a 3}})$
3. $P({\text{draw a }}\heartsuit|{\text{draw a 2}})$
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 $♠$.
54 .
Draw 5 cards without replacement from a standard deck and count the number of $♠$.
55 .
Draw cards from a standard deck and count how many cards are chosen before the first $♠$ 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 .
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 .
67 .
Which game would be better to play? Why?
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