STATS Modeling The World

David E. Bock, Paul F. Velleman, Richard D. De Veaux

Chapter 3

Gathering Data

Educators

CC

Problem 1

For each of the following scenarios, decide if the outcome is random.
a) Flip a coin to decide who takes out the trash. Is who takes out the trash random?
b) A friend asks you to quickly name a professional sports team. Is the sports team named random?
c) Names are selected out of a hat to decide roommates in a dormitory. Is your roommate for the year random?

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Problem 2

For each of the following scenarios, decide if the outcome is random.
a) You enter a contest in which the winning ticket is selected from a large drum of entries. Was the winner of the contest random?
b) When playing a board game, the number of spaces you move is decided by rolling a six-sided die. Is the number of spaces you move random?
c) Before flipping a coin, your friend asks you to “call it.” Is your choice (heads or tails) random?

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Problem 3

Many states run lotteries, giving away millions of dollars if you match a certain set of winning numbers. How are those numbers determined? Do you think this method guarantees randomness? Explain.

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Problem 4

Many kinds of games people play rely on randomness. Cite three different methods commonly used in the attempt to achieve this randomness, and discuss the effectiveness of each.

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Problem 5

The American College of Obstetricians and Gynecologists says that out of every 100 babies born in the United States, 3 have some kind of major birth defect. How would you assign random numbers to conduct a simulation based on this statistic?

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Problem 6

By some estimates, about 10% of all males have some color perception defect, most commonly red–green colorblindness. How would you assign random numbers to conduct a simulation based on this statistic?

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Problem 7

An elementary school teacher with 25 students plans to have each of them make a poster about two different states. The teacher first numbers the states (in alphabetical order, from 01-Alabama to 50-Wyoming), then uses a random number table to decide which states each kid gets. Here are the random digits:
$$45921017102289237076$$
a) Which two state numbers does the first student get?
b) Which two state numbers go to the second student?

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Problem 8

Your state’s BigBucks Lottery prize has reached $\$ 100,000,000,$ and you decide to play. You have to pick five numbers between 1 and 60, and you'll win if your numbers match those drawn by the state. You decide to pick your "lucky'r numbers using a random number table. Which numbers do you play, based on these random digits?
$$4368098750130927656158712$$

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Problem 9

Some people play state-run lotteries by always playing the same favorite “lucky” number. Assuming that the lottery is truly random, is this strategy better, worse, or the same as choosing different numbers for each play? Explain.

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Problem 10

In Exercise 8 you imagined playing the lottery by using random digits to decide what numbers to play. Is this a particularly good or bad strategy? Explain.

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Problem 11

Explain why each of the following simulations fails to model the real situation properly:
a) Use a random integer from 0 through 9 to represent the number of heads when 9 coins are tossed.
b) A basketball player takes a foul shot. Look at a random digit, using an odd digit to represent a good shot and an even digit to represent a miss.
c) Use random numbers from 1 through 13 to represent the denominations of the cards in a five-card poker hand.

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Problem 12

Explain why each of the following simulations fails to model the real situation:
a) Use random numbers 2 through 12 to represent the sum of the faces when two dice are rolled.
b) Use a random integer from 0 through 5 to represent the number of boys in a family of 5 children.
c) Simulate a baseball player's performance at bat by letting $0=$ an out, $1=$ a single, $2=$ a double, $3=$ a triple, and $4=$ a home run.

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Problem 13

A Statistics student properly simulated the length of checkout lines in a grocery store and then reported, “The average length of the line will be 3.2 people.” What’s wrong with this conclusion?

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Problem 14

After simulating the spread of a disease, a researcher wrote, “24% of the people contracted the disease.” What should the correct conclusion be?

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Problem 15

You’re pretty sure that your candidate for class president has about 55% of the votes in the entire school. But you’re worried that only 100 students will show up to vote. How often will the underdog (the one with 45% support) win? To find out, you set up a simulation.
a) Describe how you will simulate a component.
b) Describe how you will simulate a trial.
c) Describe the response variable.

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Problem 16

When drawing five cards randomly from a deck, which is more likely, two pairs or three of a kind? A pair is exactly two of the same denomination. Three of a kind is exactly 3 of the same denomination. (Don’t count three 8’s as a pair—that’s 3 of a kind. And don’t count 4 of the same kind as two pair—that’s 4 of a kind, a very special hand.) How could you simulate 5-card hands? Be careful; once you’ve picked the 8 of spades, you can’t get it again in that hand.
a) Describe how you will simulate a component.
b) Describe how you will simulate a trial.
c) Describe the response variable.

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Problem 17

In the chapter’s example, 20% of the cereal boxes contained a picture of LeBron James, 30% Danica Patrick, and the rest Serena Williams. Suppose you buy five boxes of cereal. Estimate the probability that you end up with a complete set of the pictures. Your simulation should have at least 20 runs.

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Problem 18

Suppose you really want the LeBron James picture. How many boxes of cereal do you need to buy to be pretty sure of getting at least one? Your simulation should use at least 10 trials.

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Problem 19

You take a quiz with 6 multiple choice questions. After you studied, you estimated that you would have about an 80% chance of getting any individual question right. What are your chances of getting them all right? Use at least 20 trials.

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Problem 20

A friend of yours who took the multiple choice quiz in Exercise 19 got all 6 questions right, but now claims to have guessed blindly on every question. If each question offered 4 possible answers, do you believe her? Explain, basing your argument on a simulation involving at least 10 trials.

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Problem 21

Many states run lotteries to raise money. A Web site advertises that it knows “how to increase YOUR chances of Winning the Lottery.” They offer several systems and criticize others as foolish. One system is called Lucky Numbers. People who play the Lucky Numbers system just pick a “lucky” number to play, but maybe some numbers are luckier than others. Let’s use a simulation to see how well this system works. To make the situation manageable, simulate a simple lottery in which a single digit from 0 to 9 is selected as the winning number. Pick a single value to bet, such as 1, and keep playing it over and over. You’ll want to run at least 100 trials. (If you can program the simulations on a computer, run several hundred. Or generalize the questions to a lottery that chooses two- or three-digit numbers—for which you’ll need thousands of trials.)
a) What proportion of the time do you expect to win?
b) Would you expect better results if you picked a “luckier” number, such as 7? (Try it if you don’t know.) Explain.

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Problem 22

The “beat the lottery” Web site discussed in Exercise 21 suggests that because lottery numbers are random, it is better to select your bet randomly. For the same simple lottery in Exercise 21 (random values from 0 to 9), generate each bet by choosing a separate random value between 0 and 9. Play many games. What proportion of the time do you win?

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Problem 23

The “beat the lottery” Web site of Exercise 21 notes that in the long run we expect each value to turn up about the same number of times. That leads to their recommended strategy. First, watch the lottery for a while, recording the winners. Then bet the value that has turned up the least, because it will need to turn up more often to even things out. If there is more than one “rarest” value, just take the lowest one (since it doesn’t matter). Simulating the simplified lottery described in Exercise 21, play many games with this system. What proportion of the time do you win?

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Problem 24

Another strategy for beating the lottery is the reverse of the system described in Exercise 23. Simulate the simplified lottery described in Exercise 21. Each time, bet the number that just turned up. The Web site suggests that this method should do worse. Does it? Play many games and see.

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Problem 25

You are about to take the road test for your driver’s license. You hear that only 34% of candidates pass the test the first time, but the percentage rises to 72% on subsequent retests.
a) Create a plan for a simulation to estimate the average number of tests drivers take in order to get a license.
b) Here are results of 100 trials of a simulation. Use these results to estimate the average number of tests drivers take in order to get a license.

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Problem 26

Late in a basketball game, the team that is behind often fouls someone in an attempt to get the ball back. Usually the opposing player will get to shoot foul shots “one and one,” meaning he gets a shot, and then a second shot only if he makes the first one. Suppose the opposing player has made 72% of his foul shots this season.
a) Create a plan for a simulation to estimate the number of points he will score in a one-and-one situation.
b) Here are the results of 100 trials of a simulation. Use these results to estimate the number of points he will score in a one-and-one situation.

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Problem 27

As in Exercise 25, assume that your chance of passing the driver’s test is 34% the first time and 72% for subsequent retests. Estimate the percentage of those tested who still do not have a driver’s license after two attempts.

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Problem 28

A person with type O-positive blood can receive blood only from other type O donors. About 44% of the U.S. population has type O blood. At a blood drive, how many potential donors do you expect to examine in order to get three units of type O blood?

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Problem 29

To attract shoppers, a supermarket runs a weekly contest that involves “scratch-off” cards. With each purchase, customers get a card with a black spot obscuring a message. When the spot is scratched away, most of the cards simply say, “Sorry—please try again.” But during the week, 100 customers will get cards that make them eligible for a drawing for free groceries. Ten of the cards say they may be worth $\$ 200,10$ others say $\$ 100,20$ may be worth $\$ 50,$ and the rest could be worth $\$ 20 .$ To register those cards, customers write their names A technology store holds a contest to attract shoppers. Once an hour, someone at checkout is chosen at random to play in the contest. Here’s how it works: An ace and four other cards are shuffled and placed face down on a table. The customer gets to turn over cards one at a time, looking for the ace. The person wins $\$ 100$ of store credit if the ace is the first card, $\$ 50$ if it is the second card, and $\$ 20, \$ 10,$ or $\$ 5$ if it is the third, fourth, or last card chosen. What is the average dollar amount of store credit given away in the contest? Estimate with a simulation. on them and put them in a barrel at the front of the store. At the end of the week the store manager draws cards at random, awarding the lucky customers free groceries in the amount specified on their card. The drawings continue until the store has given away more than $\$ 500$ of free groceries. Estimate the average number of winners each week.

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Problem 30

A technology store holds a contest to attract shoppers. Once an hour, someone at checkout is chosen at random to play in the contest. Here’s how it works: An ace and four other cards are shuffled and placed face down on a table. The customer gets to turn over cards one at a time, looking for the ace. The person wins $\$ 100$ of store credit if the ace is the first card, $\$ 50$ if it is the second card, and $\$ 20, \$ 10,$ or $\$ 5$ if it is the third, fourth, or last card chosen. What is the average dollar amount of store credit given away in the contest? Estimate with a simulation.

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Problem 31

Many couples want to have both a boy and a girl. If they decide to continue to have children until they have one child of each sex, what would the average family size be? Assume that boys and girls are equally likely.

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Problem 32

Suppose a couple will continue having children until they have at least two children of each sex (two boys and two girls). How many children might they expect to have?

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Problem 33

You are playing a children’s game in which the number of spaces you get to move is determined by the rolling of a die. You must land exactly on the final space in order to win. If you are 10 spaces away, how many turns might it take you to win?

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Problem 34

You are three spaces from a win in Parcheesi. On each turn, you will roll two dice. To win, you must roll a total of 3 or roll a 3 on one of the dice. How many turns might you expect this to take?

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Problem 35

A basketball player with a 65% shooting percentage has just made 6 shots in a row. The announcer
says this player “is hot tonight! She’s in the zone!” Assume the player takes about 20 shots per game. Is it unusual for her to make 6 or more shots in a row during a game?

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Problem 36

The World Series ends when a team wins 4 games. Suppose that sports analysts consider one team a bit stronger, with a 55% chance to win any individual game. Estimate the likelihood that the underdog wins the series.

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Problem 37

Four couples at a dinner party play a board game after the meal. They decide to play as teams of two and to select the teams randomly. All eight people write their names on slips of paper. The slips are thoroughly mixed, then drawn two at a time. How likely is it that every person will be teamed with someone other than the person he or she came to the party with?

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Problem 38

Suppose the couples in Exercise 37 choose the teams by having one member of each couple write their names on the cards and the other people each pick a card at random. How likely is it that every person will be teamed with someone other than the person he or she came with?

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Problem 39

A company with a large sales staff announces openings for three positions as regional managers. Twenty-two of the current salespersons apply, 12 men and 10 women. After the interviews, when the company announces the newly appointed managers, all three positions go to women. The men complain of job discrimination. Do they have a case? Simulate a random selection of three people from the applicant pool, and make a decision about the likelihood that a fair process would result in hiring all women.

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Charles C.
Numerade Educator

Problem 40

A proud legislator claims that your state’s new law banning texting and hand-held phones while driving reduced occurrences of infractions to less than 10% of all drivers. While on a long drive home from your college, you notice a few people seemingly texting. You decide to count everyone using their smartphones illegally who pass you on the expressway for the next 20 minutes. It turns out that 5 out of the 20 drivers were actually using their phones illegally. Does this cast doubt on the legislator’s figure of 10%? Use a simulation to estimate the likelihood of seeing at least 5 out of 20 drivers using their phones illegally if the actual usage rate is only 10%. Explain your conclusion clearly.

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