7.12.4: Chapter Review
The Multiplication Rule for Counting
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?
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?
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?
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
Compute \(\frac{8!}{2!3!3!}\).
Compute \(\frac{12!}{8!3!}\).
Compute \(\frac{211!}{210!}\)
Compute \(_5{P_3}\).
Compute \(_{15}{P_3}\).
Compute \(_{22}{P_5}\).
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?
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
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?
If you’re trying to decide which of your guests sits where at your table, are you using permutations or combinations?
Compute \(_7{C_4}\).
Compute \(_{13}{C_8}\).
How many ways are there to draw a hand of 8 cards from a deck of 16 cards?
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
If you draw a card at random from a standard 52-card deck and note its suit, what is the sample space?
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?
If you draw 2 Scrabble tiles in order without replacement from a bag containing E, E, L, S, what is the sample space?
If you draw 2 Scrabble tiles without replacement and ignoring order from a bag containing E, E, L, S, what is the sample space?
If you draw 2 Scrabble tiles without replacement and ignoring order from a bag containing E, E, L, S, what is the sample space?
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
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?
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?
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).
What is the probability of rolling a negative number?
What is the probability of rolling a number less than 20?
What is the probability of rolling an 11?
What is the probability of rolling a number less than 7?
What is the probability of not rolling an 11?
What is the probability of rolling a multiple of 4?
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 \(2♡\), \(2\clubsuit\), \(2\diamondsuit\), \(2\spadesuit\), \(3♡\), \(3\clubsuit\), \(3\spadesuit\), \(4♡\), \(4♣\), \(5♡\).
If you draw 4 cards without replacement, what is the probability of drawing a 2, 3, 4, and 5 in order?
If you draw 4 cards without replacement, what is the probability of drawing a 2, 3, 4, and 5 in any order?
If you draw 3 cards without replacement, what is the probability that you draw a \(♡\), a \(♣\), and a \(♡\), in order?
If you draw 3 cards without replacement, what is the probability that you draw 2 \(♡\) and 1 \(♣\), in any order?
What Are the Odds?
If you roll a standard 20-sided die (with faces numbered 1–20), what are the odds against rolling a number less than 5?
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?
If \(P(E) = \frac{4}{7}\), what are the odds in favor of \(E\)?
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 \(2♡\), \(2♣\), \(2\diamondsuit\), \(2♠\), \(3♡\), \(3♣\), \(3♠\), \(4♡\), \(4♣\), \(5♡\).
What is the probability of drawing a 2 or a 3?
What is the probability of drawing a \(♣\) or a \(♠\)?
What is the probability of drawing a 2 or a \(♡\)?
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 \(2♡\), \(2♣\), \(2\diamondsuit\), \(2♠\), \(3♡\), \(3♣\), \(3♠\), \(4♡\), \(4♣\), \(5♡\).
If you draw a single card, what is:
- \(P({\text{draw a 2}})\)
- \(P({\text{draw a 2}}|{\text{draw a }}\heartsuit)\)
- \(P({\text{draw a 2}}|{\text{draw a }}\spadesuit )\)
If you draw a single card, what is:
- \(P({\text{draw a }}\heartsuit)\)
- \(P({\text{draw a }}\heartsuit|{\text{draw a 3}})\)
- \(P({\text{draw a }}\heartsuit|{\text{draw a 2}})\)
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:
\(P({\text{first roll is a 3 and second roll is a 3}})\)
\(P({\text{first roll is a 6 and second roll is a 6}})\)
\(P({\text{second roll is a 6}})\)
\(P({\text{second roll is a 1}})\)
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.
Draw 5 cards with replacement from a standard deck and count the number of \(♠\).
Draw 5 cards without replacement from a standard deck and count the number of \(♠\).
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.
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.
If you roll the die 10 times, what is the probability that a number between 1 and 5 (inclusive) comes up exactly once?
If you roll the die 40 times, what is the probability that 20 comes up fewer than 2 times?
If you roll the die 40 times, what is the probability that 20 comes up 4 or more times?
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)?
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?
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
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?
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.
What is the expected value of this game?
Interpret your answer.
Which game would be better to play? Why?