The Practice of Statistics for AP

Daren S. Starnes, Daniel S. Yates, David S. Moore

Chapter 5

Probability: What Are the Chances? - all with Video Answers

Educators

Section 1

Randomness, Probability, and Simulation

Problem 1

Liar, liar! Sometimes police use a lie detector (also known as a polygraph) to help determine whether
a suspect is telling the truth. A lie detector test isn’t foolproof—sometimes it suggests that a person is lying when they’re actually telling the truth (a false positive). Other times, the test says that the suspect is being truthful when the person is actually lying (a false negative). For one brand of polygraph machine, the probability of a false positive is 0.08. (a) Interpret this probability as a long-run relative frequency. (b) Which is a more serious error in this case: a false positive or a false negative? Justify your answer.

Alice Simper

Alice Simper

Numerade Educator

Problem 2

Drug testing Athletes are often tested for use of performance-enhancing drugs. Drug tests aren’t
perfect—they sometimes say that an athlete took a banned substance when that isn’t the case (a false positive). Other times, the test concludes that the athlete is clean when he or she actually took a banned substance (a false negative). For one commonly used drug test, the probability of a false negative is 0.03. (a) Interpret this probability as a long-run relative frequency. (b) Which is a more serious error in this case: a false positive or a false negative? Justify your answer.

Alice Simper

Numerade Educator

Problem 3

Genetics Suppose a married man and woman both carry a gene for cystic fibrosis but don’t have the disease themselves. According to the laws of genetics, the probability that their first child will develop cystic fibrosis is 0.25. (a) Explain what this probability means. (b) Why doesn’t this probability say that if the couple has 4 children, one of them is guaranteed to get cystic fibrosis?

Alice Simper

Numerade Educator

Problem 4

Texas hold ’em In the popular Texas hold ’em variety of poker, players make their best five-card poker
hand by combining the two cards they are dealt with three of five cards available to all players. You read in a book on poker that if you hold a pair (two cards of the same rank) in your hand, the probability of getting four of a kind is 88/1000. (a) Explain what this probability means. (b) Why doesn’t this probability say that if you play 1000 such hands, exactly 88 will be four of a kind?

Alice Simper

Numerade Educator

Problem 5

Spinning a quarter With your forefinger, hold a new quarter (with a state featured on the reverse) upright,
on its edge, on a hard surface. Then flick it with your other forefinger so that it spins for some time before it falls and comes to rest. Spin the coin a total of 25 times, and record the results. (a) What’s your estimate for the probability of heads? Why? (b) Explain how you could get an even better estimate.

Alice Simper

Numerade Educator

Problem 6

Nickels falling over You may feel it’s obvious that the probability of a head in tossing a coin is about 1/2
because the coin has two faces. Such opinions are not always correct. Stand a nickel on edge on a hard, flat surface. Pound the surface with your hand so that the nickel falls over. Do this 25 times, and record the results. (a) What’s your estimate for the probability that the coin falls heads up? Why? (b) Explain how you could get an even better estimate.

Alice Simper

Numerade Educator

Problem 7

Free throws The figure below shows the results of a basketball player shooting several free throws. Explain what this graph says about chance behavior in the short run and long run.

Alice Simper

Numerade Educator

Problem 8

Keep on tossing The figure below shows the results of two different sets of 5000 coin tosses. Explain what this graph says about chance behavior in the short run and the long run.

Alice Simper

Numerade Educator

Problem 9

Due for a hit A very good professional baseball player gets a hit about 35% of the time over an entire
season. After the player failed to hit safely in six straight at-bats, a TV commentator said, He is due
for a hit by the law of averages. Is that right? Why?

Alice Simper

Numerade Educator

Problem 10

Cold weather coming A TV weather man, predicting a colder-than-normal winter, said, First, in looking at the past few winters, there has been a lack of really cold weather. Even though we are not supposed to use the law of averages, we are due. Do you think that due by the law of averages makes sense in talking about the weather? Why or why not?

Alice Simper

Numerade Educator

Problem 11

Playing Pick 4 The Pick 4 games in many state lotteries announce a four-digit winning number each
day. You can think of the winning number as a four-digit group from a table of random digits. You win (or share) the jackpot if your choice matches the winning number. The winnings are divided among all players who matched the winning number. That suggests a way to get an edge. (a) The winning number might be, for example, either 2873 or 9999. Explain why these two outcomes have exactly the same probability. (b) If you asked many people whether 2873 or 9999 is more likely to be the randomly chosen winning number, most would favor one of them. Use the information in this section to say which one and to explain why. How might this affect the four-digit number you would choose?

Alice Simper

Numerade Educator

Problem 12

An unenlightened gambler (a) A gambler knows that red and black are equally likely to occur on each spin of a roulette wheel. He observes five consecutive reds occur and bets heavily on black at the next spin. Asked why, he explains that black is due by the law of averages. Explain to the gambler what is wrong with this reasoning. (b) After hearing you explain why red and black are still equally likely after five reds on the roulette wheel, the gambler moves to a poker game. He is dealt five straight red cards. He remembers what you said and assumes that the next card dealt in the same hand is equally likely to be red or black. Is the gambler right or wrong, and why?

Alice Simper

Numerade Educator

Problem 13

Free throws A basketball player has probability 0.75 of making a free throw. Explain how you would use
each chance device to simulate one free throw by the player.
(a) A six-sided die
(b) Table D of random digits
(c) A standard deck of playing cards

Alice Simper

Numerade Educator

Problem 14

Stoplight On her drive to work every day, Ilana passes through an intersection with a traffic light. The
light has probability 1/3 of being green when she gets to the intersection. Explain how you would use each chance device to simulate whether the light is red or green on a given day.
(a) A six-sided die
(b) Table D of random digits
(c) A standard deck of playing cards

Alice Simper

Numerade Educator

Problem 15

Simulation blunders Explain what’s wrong with each of the following simulation designs. (a) A roulette wheel has 38 colored slots—18 red, 18 black, and 2 green. To simulate one spin of the wheel, let numbers 00 to 18 represent red, 19 to 37 represent black, and 38 to 40 represent green. (b) About 10% of U.S. adults are left-handed. To simulate randomly selecting one adult at a time until you find a left-hander, use two digits. Let 01 to 10 represent being left-handed and 11 to 00 represent being right-handed. Move across a row in Table D, two digits at a time, skipping any numbers that have already appeared, until you find a number between 01 and 10. Record the number of people selected.

Alice Simper

Numerade Educator

Problem 16

Simulation blunders Explain what’s wrong with each of the following simulation designs. (a) According to the Centers for Disease Control and Prevention, about 26% of U.S. adults were obese in 2008. To simulate choosing 8 adults at random and seeing how many are obese, we could use two digits. Let 01 to 26 represent obese and 27 to 00 represent not obese. Move across a row in Table D, two digits at a time, until you find 8 distinct numbers (no repeats). Record the number of obese people selected. (b) Assume that the probability of a newborn being a boy is 0.5. To simulate choosing a random sample of 9 babies who were born at a local hospital today and observing their gender, use one digit. Use randint (0,9) on your calculator to determine how many babies in the sample are male.

Sheryl Ezze

Numerade Educator

Problem 17

Is this valid? Determine whether each of the following simulation designs is valid. Justify your answer. (a) According to a recent poll, 75% of American adults regularly recycle. To simulate choosing a random sample of 100 U.S. adults and seeing how many of them recycle, roll a 4-sided die 100 times. A result of 1, 2, or 3 means the person recycles; a 4 means that the person doesn’t recycle. (b) An archer hits the center of the target with 60% of her shots. To simulate having her shoot 10 times, use a coin. Flip the coin once for each of the 10 shots. If it lands heads, then she hits the center of the target. If the coin lands tails, she doesn’t.

Alice Simper

Numerade Educator

Problem 18

Is this valid? Determine whether each of the following simulation designs is valid. Justify your answer. (a) According to a recent survey, 50% of people aged 13 and older in the United States are addicted to email. To simulate choosing a random sample of 20 people in this population and seeing how many of them are addicted to email, use a deck of cards. Shuffle the deck well, and then draw one card at a time. A red card means that person is addicted to email; a black card means he isn’t. Continue until you have drawn 20 cards (without replacement) for the sample. (b) A tennis player gets 95% of his second serves in play during practice (that is, the ball doesn’t go out of bounds). To simulate the player hitting 5 second serves, look at pairs of digits going across a row in Table D. If the number is between 00 and 94, the serve is in; numbers between 95 and 99 indicate that the serve is out.

Alice Simper

Numerade Educator

Problem 19

Airport security The Transportation Security Administration (TSA) is responsible for airport safety. On some flights, TSA officers randomly select passengers for an extra security check prior to boarding. One such flight had 76 passengers—12 in first class and 64 in coach class. Some passengers were surprised when none of the 10 passengers chosen for screening were seated in first class. We can use a simulation to see if this result is likely to happen by chance. (a) State the question of interest using the language of probability. (b) How would you use random digits to imitate one repetition of the process? What variable would you measure? (c) Use the line of random digits below to perform one repetition. Copy these digits onto your paper. Mark directly on or above them to show how you determined the outcomes of the chance process.
71487 09984 29077 14863 61683 47052 62224 51025
(d) In 100 repetitions of the simulation, there were 15 times when none of the 10 passengers chosen
was seated in first class. What conclusion would you draw?

Alice Simper

Numerade Educator

Problem 20

Scrabble In the game of Scrabble, each player begins by drawing 7 tiles from a bag containing 100 tiles. There are 42 vowels, 56 consonants, and 2 blank tiles in the bag. Cait chooses her 7 tiles and is surprised to discover that all of them are vowels. We can use a simulation to see if this result is likely to happen by chance. (a) State the question of interest using the language of probability. (b) How would you use random digits to imitate one repetition of the process? What variable would you measure? (c) Use the line of random digits below to perform one repetition. Copy these digits onto your paper. Mark directly on or above them to show how you determined the outcomes of the chance process.
00694 05977 19664 65441 20903 62371 22725 53340 (d) In 1000 repetitions of the simulation, there were 2 times when all 7 tiles were vowels. What conclusion would you draw?

Alice Simper

Numerade Educator

Problem 21

The birthday problem What’s the probability that in a randomly selected group of 30 unrelated people, at least two have the same birthday? Let’s make two assumptions to simplify the problem. First, well ignore the possibility of a February 29 birthday. Second, we assume that a randomly chosen person is equally likely to be born on each of the remaining 365 days of the year. (a) How would you use random digits to imitate one repetition of the process? What variable would you measure? (b) Use technology to perform 5 repetitions. Record the outcome of each repetition. (c) Would you be surprised to learn that the theoretical probability is 0.71? Why or why not?

Alice Simper

Numerade Educator

Problem 22

Monty Hall problem In Parade magazine, a reader posed the following question to Marilyn vos Savant and the Ask Marilyn column: Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, Do you want to pick door #2? Is it to your advantage to switch your choice of doors? The game show in question was Let’s Make a Deal and the host was Monty Hall. Here’s the first part of Marilyn’s response: Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Thousands of readers wrote to Marilyn to disagree with her answer. But she held her ground. (a) Use an online Let’s Make a Deal applet to perform at least 50 repetitions of the simulation. Record whether you stay or switch (try to do each about half the time) and the outcome of each repetition. (b) Do you agree with Marilyn or her readers? Explain.

Sheryl Ezze

Numerade Educator

Problem 23

Recycling Do most teens recycle? To find out, an AP Statistics class asked an SRS of 100 students at their school whether they regularly recycle. How many students in the sample would need to say Yes to provide convincing evidence that more than half of the students at the school recycle? The Fathom dotplot below shows the results of taking 200 SRSs of 100 students from a population in which the true proportion who recycle is 0.50. (a) Suppose 55 students in the class’s sample say Yes. Explain why this result does not give convincing evidence that more than half of the school’s students recycle. (b) Suppose 63 students in the class’s sample say Yes. Explain why this result gives strong evidence that a majority of the school’s students recycle.

Dominador Tan

Numerade Educator

Problem 24

Brushing teeth, wasting water? A recent study reported that fewer than half of young adults turn
off the water while brushing their teeth. Is the same true for teenagers? To find out, a group of statistics students asked an SRS of 60 students at their school if they usually brush with the water off. How many students in the sample would need to say No to provide convincing evidence that fewer than half of the students at the school brush with the water off? The Fathom dotplot below shows the results of taking 200 SRSs of 60 students from a population in which the true proportion who brush with the water off is 0.50. (a) Suppose 27 students in the class’s sample say No. Explain why this result does not give convincing evidence that fewer than half of the school’s students brush their teeth with the water off. (b) Suppose 18 students in the class’s sample say No. Explain why this result gives strong evidence that fewer than 50% of the school’s students brush their teeth with the water off.

Alice Simper

Numerade Educator

Problem 25

Color-blind men About 7% of men in the United States have some form of red-green color blindness. Suppose we randomly select one U.S. adult male at a time until we find one who is red-green color-blind. How many men would we expect to choose, on average? Design and carry out a simulation to answer this question. Follow the four-step process.

Alice Simper

Numerade Educator

Problem 26

Scrabble Refer to Exercise 20. About 3% of the time, the first player in Scrabble can bingo by playing all 7 tiles on the first turn. How many games of Scrabble would you expect to have to play, on average, for this to happen? Design and carry out a simulation to answer this question. Follow the four-step process.

Alice Simper

Numerade Educator

Problem 27

Color-blind men Refer to Exercise 25. Suppose we randomly select 4 U.S. adult males. What’s the
probability that at least one of them is red-green color-blind? Design and carry out a simulation to
answer this question. Follow the four-step process.

Alice Simper

Numerade Educator

Problem 28

Lotto In the United Kingdom’s Lotto game, a player picks six numbers from 1 to 49 for each ticket. Rosemary bought one ticket for herself and one for each of her four adult children. She had the lottery computer randomly select the six numbers on each ticket. When the six winning numbers were drawn, Rosemary was surprised to find that none of these numbers appeared on any of the five Lotto tickets she had bought. Should she be? Design and carry out a simulation to answer this question. Follow the four-step process.

Alice Simper

Numerade Educator

Problem 29

Random assignment Researchers recruited 20 volunteers—8 men and 12 women—to take part in an experiment. They randomly assigned the subjects into two groups of 10 people each. To their surprise, 6 of the 8 men were randomly assigned to the same treatment. Should they be surprised? Design and carry out a simulation to estimate the probability that the random assignment puts 6 or more men in the same group. Follow the four-step process.

Alice Simper

Numerade Educator

Problem 30

Taking the train According to New Jersey Transit, the 8:00 a.m. weekday train from Princeton
to New York City has a 90% chance of arriving on time. To test this claim, an auditor chooses 6
weekdays at random during a month to ride this train. The train arrives late on 2 of those days.
Does the auditor have convincing evidence that the company’s claim isn’t true? Design and
carry out a simulation to answer this question. Follow the four-step process.

Alice Simper

Numerade Educator

Problem 31

You read in a book about bridge that the probability that each of the four players is dealt exactly one ace is about 0.11. This means that (a) in every 100 bridge deals, each player has one ace exactly 11 times.
(b) in one million bridge deals, the number of deals on which each player has one ace will be exactly
110,000. (c) in a very large number of bridge deals, the percent of deals on which each player has one ace will be very close to 11%.(d) in a very large number of bridge deals, the average number of aces in a hand will be very close to 0.11. (e) None of these

Alice Simper

Numerade Educator

Problem 32

If 1 toss a fair coin five times and the outcomes are TTTTTT, then the probability that tails appears on the next toss is
(a) 0.5.
(b) less than 0.5.
(c) greater than 0.5.
(d) 0
(e) 1.

Alice Simper

Numerade Educator

Problem 33

Exercises 33 to 35 refer to the following setting. A basketball player makes 47% of her shots from the field during the season.
To simulate whether a shot hits or misses, you would assign random digits as follows:
(a) One digit simulates one shot; 4 and 7 are a hit; other digits are a miss.
(b) One digit simulates one shot; odd digits are a hit and even digits are a miss.
(c) Two digits simulate one shot; 00 to 47 are a hit and 48 to 99 are a miss.
(d) Two digits simulate one shot; 00 to 46 are a hit and 47 to 99 are a miss.
(e) Two digits simulate one shot; 00 to 45 are a hit and 46 to 99 are a miss.

Alice Simper

Numerade Educator

Problem 34

Exercises 33 to 35 refer to the following setting. A basketball player makes 47% of her shots from the field during the season.
Use the correct choice from the previous question and these random digits to simulate 10 shots:
82734 71490 20467 47511 81676 55300 94383 14893
How many of these 10 shots are hits?
(a) 2 (b) 3 (c) 4 (d) 5 (e) 6

Alice Simper

Numerade Educator

Problem 35

Exercises 33 to 35 refer to the following setting. A basketball player makes 47% of her shots from the field during the season.
You want to estimate the probability that the player makes 5 or more of 10 shots. You simulate 10 shots
25 times and get the following numbers of hits:
5754153434534463417455657
What is your estimate of the probability?
(a) 5/25, or 0.20
(b) 11/25, or 0.44
(c) 12/25, or 0.48
(d) 16/25, or 0.64
(e) 19/25, or 0.76

Alice Simper

Numerade Educator

Problem 36

Ten percent of U.S. households contain 5 or more people. You want to simulate choosing a household at
random and recording whether or not it contains 5 or more people. Which of these are correct assignments of digits for this simulation?
(a) Odd = Yes (5 or more people); Even = No (not 5 or more people)
(b) 0 = Yes; 1, 2, 3, 4, 5, 6, 7, 8, 9 = No
(c) 5 = Yes; 0, 1, 2, 3, 4, 6, 7, 8, 9 = No
(d) All three are correct.
(e) Choices (b) and (c) are correct, but (a) is not.

Alice Simper

Numerade Educator

Problem 37

Are you feeling stressed? (4.1) A Gallup Poll asked whether people experienced stress a lot of the day yesterday. Forty percent said they did. Gallup’s report said, “Results are based on telephone interviews with 178,545 national adults, aged 18 and older, conducted Jan. 2–June 30, 2009$^{4}$
(a) Identify the population and the sample.
(b) Explain how undercoverage could lead to bias in this survey.

Alice Simper

Numerade Educator

Problem 38

Waiting to park (1.3) Do drivers take longer to leave their parking spaces when someone is waiting? Researchers hung out in a parking lot and collected some data. The graphs and numerical summaries below display information about how long it took drivers to exit their places.
(a) Write a few sentences comparing these distributions.
(b) Can we conclude that having someone waiting causes drivers to leave their spaces more slowly? Why
or why not?

Alice Simper

Numerade Educator