Stats Data and Models

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

Chapter 16

Probability Models - all with Video Answers

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Chapter Questions

Problem 1

Bernoulli Do these situations involve Bernoulli trials? Explain.
a) We roll 50 dice to find the distribution of the number of spots on the faces.
b) How likely is it that in a group of 120 the majority may have Type A blood, given that Type $A$ is found in $43 \%$ of the population?
c) We deal 7 cards from a deck and get all hearts. How likely is that?
d) We wish to predict the outcome of a vote on the school budget, and poll 500 of the 3000 likely voters to see how many favor the proposed budget.
e) A company realizes that about $10 \%$ of its packages are not being sealed properly. In a case of 24 , is it likely that more than 3 are unsealed?

James Kiss

James Kiss

Numerade Educator

Problem 2

Bernoulli 2 Do these situations involve Bernoulli trials? Explain.
a) You are rolling 5 dice and need to get at least two 6 's to win the game.
b) We record the distribution of eye colors found in a group of 500 people.
c) A manufacturer recalls a doll because about $3 \%$ have buttons that are not properly attached. Customers return 37 of these dolls to the local toy store. Is the manufacturer likely to find any dangerous buttons?
d) A city council of 11 Republicans and 8 Democrats picks a committee of 4 at random. What's the probability they choose all Democrats?
e) A 2002 Rutgers University study found that $74 \%$ of high school students have cheated on a test at least once. Your local high-school principal conducts a survey in homerooms and gets responses that admit to cheating from 322 of the 481 students.

James Kiss

Numerade Educator

Problem 3

Toasters A manufacturer ships toasters in cartons of 20 . In each carton, they estimate a $5 \%$ chance that one of the toasters will need to be sent back for minor repairs. What is the probability that in a carton, there will be exactly 3 toasters that need repair?

James Kiss

Numerade Educator

Problem 4

Soccer A soccer team estimates that they will score on $8 \%$ of the corner kicks. In next week's game, the team hopes to kick 15 corner kicks. What are the chances that they will score on 2 of those opportunities?

James Kiss

Numerade Educator

Problem 5

Toasters again In a batch of 10,000 toasters, what are the chances that fewer than 450 need to be returned?

James Kiss

Numerade Educator

Problem 6

Soccer again If this team has 200 corner kicks over the season, what are the chances that they score more than 22 times?

James Kiss

Numerade Educator

Problem 7

Sell! A car dealership sells an average of 5 cars in a day. Using the Poisson model, what is the probability that the dealer sells 3 cars tomorrow?

James Kiss

Numerade Educator

Problem 8

Passing on A large hospital has an average of 7 fatalities in a week. Using the Poisson model, what is the probability that this week it has 10 fatalities?

James Kiss

Numerade Educator

Problem 9

Telephone numbers A cable provider wants to contact customers in a particular telephone exchange to see how satisfied they are with the new digital TV service the company has provided. All numbers are in the 452 exchange, so there are 10,000 possible numbers from $452-0000$ to $452-9999 .$ If they select the numbers with equal probability:
a) What distribution would they use to model the selection?
b) The new business "incubator" was assigned the 200 numbers between $452-2500$ and $452-2699,$ but these businesses don't subscribe to digital TV. What is the probability that the randomly selected number will be for an incubator business?
c) Numbers above 9000 were only released for domestic use last year, so they went to newly constructed residences. What is the probability that a randomly selected number will be one of these?

James Kiss

Numerade Educator

Problem 10

Serial numbers In an effort to check the quality of their cell phones, a manufacturing manager decides to take a random sample of 10 cell phones from yesterday's production run, which produced cell phones with serial numbers ranging (according to when they were produced) from 43005000 to $43005999 .$ If each of the 1000 phones is equally likely to be selected:
a) What distribution would they use to model the selection?
b) What is the probability that a randomly selected cell phone will be one of the last 100 to be produced?
c) What is the probability that the first cell phone selected is either from the last 200 to be produced or from the first 50 to be produced?

James Kiss

Numerade Educator

Problem 11

Component lifetimes Lifetimes of electronic components can often be modeled by an exponential model. Suppose quality control engineers want to model the lifetime of a hard drive to have a mean lifetime of 3 years.
a) What value of $\lambda$ should they use?
b) With this model, what would the probability be that a hard drive lasts 5 years or less?

James Kiss

Numerade Educator

Problem 12

Website sales Suppose occurrences of sales on a small company's website are well modeled by a Poisson model with $\lambda=5 /$ hour.
a) If a sale just occurred, what is the expected waiting time until the next sale?
b) What is the probability that the next sale will happen in the next 6 minutes?

James Kiss

Numerade Educator

Problem 13

Simulating the model Think about the Hope Solo picture search again. You are opening boxes of cereal one at a time looking for her picture, which is in $20 \%$ of the boxes. You want to know how many boxes you might have to open in order to find Hope.
a) Describe how you would simulate the search for Hope using random numbers.
b) Run at least 30 trials.
c) Based on your simulation, estimate the probabilities that you might find your first picture of Hope in the first box, the second, etc.
d) Calculate the actual probability model.
e) Compare the distribution of outcomes in your simulation to the probability model.

Sheryl Ezze

Numerade Educator

Problem 14

Simulation II You are one space short of winning a child's board game and must roll a 1 on a die to claim victory. You want to know how many rolls it might take.
a) Describe how you would simulate rolling the die until you get a 1 .
b) Run at least 30 trials.
c) Based on your simulation, estimate the probabilities that you might win on the first roll, the second, the third, etc.
d) Calculate the actual probability model.
e) Compare the distribution of outcomes in your simulation to the probability model.

Sheryl Ezze

Numerade Educator

Problem 15

Hope, again Let's take one last look at the Hope Solo picture search. You know her picture is in $20 \%$ of the cereal boxes. You buy five boxes to see how many pictures of Hope you might get.
a) Describe how you would simulate the number of pictures of Hope you might find in five boxes of cereal.
b) Run at least 30 trials.
c) Based on your simulation, estimate the probabilities that you get no pictures of Hope, 1 picture, 2 pictures, etc.
d) Find the actual probability model.
e) Compare the distribution of outcomes in your simulation to the probability model.

Sheryl Ezze

Numerade Educator

Problem 16

Seatbelts Suppose $75 \%$ of all drivers always wear their seatbelts. Let's investigate how many of the drivers might be belted among five cars waiting at a traffic light.
a) Describe how you would simulate the number of seatbelt-wearing drivers among the five cars.
b) Run at least 30 trials.
c) Based on your simulation, estimate the probabilities there are no belted drivers, exactly one, two, etc.
d) Find the actual probability model.
e) Compare the distribution of outcomes in your simulation to the probability model.

James Kiss

Numerade Educator

Problem 17

On time A Department of Transportation report about air travel found that, nationwide, $76 \%$ of all flights are on time. Suppose you are at the airport and your flight is one of 50 scheduled to take off in the next two hours. Can you consider these departures to be Bernoulli trials? Explain.

Robin Corrigan

Numerade Educator

Problem 18

Lost luggage A Department of Transportation report about air travel found that airlines misplace about 5 bags per 1000 passengers. Suppose you are traveling with a group of people who have checked 22 pieces of luggage on your flight. Can you consider the fate of these bags to be Bernoulli trials? Explain.

James Kiss

Numerade Educator

Problem 19

Hoops A basketball player has made $60 \%$ of his foul shots during the season. Assuming the shots are independent, find the probability that in tonight's game he
a) misses for the first time on his fifth attempt.
b) makes his first basket on his fourth shot.
c) makes his first basket on one of his first 3 shots.

Sanchit Jain

Numerade Educator

Problem 20

Chips Suppose a computer chip manufacturer rejects $2 \%$ of the chips produced because they fail presale testing.
a) What's the probability that the fifth chip you test is the first bad one you find?
b) What's the probability you find a bad one within the first 9 you examine?

Sanchit Jain

Numerade Educator

Problem 21

More hoops For the basketball player in Exercise 19 , what's the expected number of shots until he misses?

Sheryl Ezze

Numerade Educator

Problem 22

Chips ahoy For the computer chips described in Exercise 20 , how many do you expect to test before finding a bad one?

Sheryl Ezze

Numerade Educator

Problem 23

Customer center operator Raaj works at the customer service call center of a major credit card bank. Cardholders call for a variety of reasons, but regardless of their reason for calling, if they hold a platinum card, Raaj is instructed to offer them a double-miles promotion. About $14 \%$ of all cardholders hold platinum cards, and about $55 \%$ of those will take the double-miles promotion. On average, how many calls will Raaj have to take before finding the first cardholder to take the double-miles promotion?

Sheryl Ezze

Numerade Educator

Problem 24

Cold calls Justine works for an organization committed to raising money for Alzheimer's research. From past experience, the organization knows that about $20 \%$ of all potential donors will agree to give something if contacted by phone. They also know that of all people donating, about $5 \%$ will give $\$ 100$ or more. On average, how many potential donors will she have to contact until she gets her first $\$ 100$ donor?

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 25

Blood Only $4 \%$ of people have Type AB blood.
a) On average, how many donors must be checked to find someone with Type AB blood?
b) What's the probability that there is a Type AB donor among the first 5 people checked?
c) What's the probability that the first Type AB donor will be found among the first 6 people?
d) What's the probability that we won't find a Type $\mathrm{AB}$ donor before the 10 th person?

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 26

Color blindness About $8 \%$ of males are color-blind. A researcher needs some color-blind subjects for an experiment and begins checking potential subjects.
a) On average, how many men should the researcher expect to check to find one who is color-blind?
b) What's the probability that she won't find anyone color-blind among the first 4 men she checks?
c) What's the probability that the first color-blind man found will be the sixth person checked?
d) What's the probability that she finds someone who is color-blind before checking the 10 th man?

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 27

Coins and intuition If you flip a fair coin 160 times,
a) Intuitively, how many heads do you expect?
b) Use the formula for expected value to verify your intuition.

James Kiss

Numerade Educator

Problem 28

Roulette and intuition An American roulette wheel has 38 slots, of which 18 are red, 18 are black, and 2 are green $(0$ and 00$)$. If you spin the wheel 38 times,
a) Intuitively, how many times would you expect the ball to wind up in a green slot?
b) Use the formula for expected value to verify your intuition.

James Kiss

Numerade Educator

Problem 29

Lefties Assume that $12 \%$ of people are left-handed. If we select 4 people at random, find the probability of each outcome.
a) The first lefty is the fourth person chosen.
b) There are some lefties among the 4 people.
c) The first lefty is the third or fourth person.
d) There are exactly 3 lefties in the group.
c) There are at least 2 leftics in the group.
f) There are no more than 2 lefties in the group.

Willis James

Numerade Educator

Problem 30

Arrows An Olympic archer is able to hit the bull's-eye $80 \%$ of the time. Assume each shot is independent of the others. If she shoots 6 arrows, what's the probability of each of the following results?
a) Her first bull's-eye comes on the third arrow.
b) She misses the bull's-eye at least once.
c) Her first bull's-eye comes on the fourth or fifth arrow.
d) She gets exactly 4 bull's-eyes.
e) She gets at least 4 bull's-eyes.
f) She gets at most 4 bull's-eyes.

Khalida Dawar

Numerade Educator

Problem 31

Lefties, redux Consider our group of 5 people from Exercise $29 .$
a) How many lefties do you expect?
b) With what standard deviation?
c) If we keep picking people until we find a lefty, how long do you expect it will take?

James Kiss

Numerade Educator

Problem 32

More arrows Consider our archer from Exercise $30 .$
a) How many bull's-eyes do you expect her to get?
b) With what standard deviation?
c) If she keeps shooting arrows until she hits the bull'seye, how long do you expect it will take?

James Kiss

Numerade Educator

Problem 33

Still more lefties Suppose we choose 10 people instead of the 5 chosen in Exercise $29 .$
a) Find the mean and standard deviation of the number of right-handers in the group.
b) What's the probability that they're not all right-handed?
c) What's the probability that there are no more than 10 righties?
d) What's the probability that there are exactly 6 of each?
e) What's the probability that the majority is right-handed?

Sheryl Ezze

Numerade Educator

Problem 34

Still more arrows Suppose the archer from Exercise 30 shoots 10 arrows.
a) Find the mean and standard deviation of the number of bull's-eyes she may get.
b) What's the probability that she never misses?
c) What's the probability that there are no more than 8 bull's-eyes?
d) What's the probability that there are exactly 8 bull's-eyes?
e) What's the probability that she hits the bull's-eye more often than she misses?

Sheryl Ezze

Numerade Educator

Problem 35

Vision It is generally believed that nearsightedness affects about $14 \%$ of all children. A school district tests the vision of 156 incoming kindergarten children. How many would you expect to be nearsighted? With what standard deviation?

James Kiss

Numerade Educator

Problem 36

International students At a certain college, $6 \%$ of all students come from outside the United States. Incoming students there are assigned at random to freshman dorms, where students live in residential clusters of 55 freshmen sharing a common lounge area. How many international students would you expect to find in a typical cluster? With what standard deviation?

James Kiss

Numerade Educator

Problem 37

Tennis, anyone? A certain tennis player makes a successful first serve $70 \%$ of the time. Assume that each serve is independent of the others. If she serves 4 times, what's the probability she gets
a) all 4 serves in?
b) exactly 3 serves in?
c) at least 2 serves in?
d) no more than 3 serves in?

Michelle Z.

Numerade Educator

Problem 38

Frogs A wildlife biologist examines frogs for a genetic trait he suspects may be linked to sensitivity to industrial toxins in the environment. Previous research had established that this trait is usually found in 1 of every 8 frogs. He collects and examines a dozen frogs. If the frequency of the trait has not changed, what's the probability he finds the trait in
a) none of the 12 frogs?
b) at least 2 frogs?
c) 3 or 4 frogs?
d) no more than 4 frogs?

Sheryl Ezze

Numerade Educator

Problem 39

And more tennis Suppose the tennis player in Exercise 37 serves 80 times in a match.
a) What are the mean and standard deviation of the number of good first serves expected?
b) Verify that you can use a Normal model to approximate the distribution of the number of good first serves.
c) Use the $68-95-99.7$ Rule to describe this distribution.
d) What's the probability she makes at least 65 first serves?

Sheryl Ezze

Numerade Educator

Problem 40

More arrows The archer in Exercise 30 will be shooting 200 arrows in a large competition.
a) What are the mean and standard deviation of the number of bull's-eyes she might get?
b) Is a Normal model appropriate here? Explain.
c) Use the $68-95-99.7$ Rule to describe the distribution of the number of bull's-eyes she may get.
d) Would you be surprised if she made only 140 bull'seyes? Explain.

Sheryl Ezze

Numerade Educator

Problem 41

Apples An orchard owner knows that he'll have to use about $5 \%$ of the apples he harvests for cider because they will have bruises or blemishes. He expects a tree to produce about 200 apples.
a) Describe an appropriate model for the number of cider apples that may come from that tree. Justify your model.
b) Find the probability there will be no more than a dozen cider apples.
c) Is it likely there will be more than 55 cider apples? Explain.

Sheryl Ezze

Numerade Educator

Problem 42

Frogs, part II Based on concerns raised by his preliminary research, the biologist in Exercise 38 decides to collect and examine 150 frogs.
a) Assuming the frequency of the trait is still 1 in 8 , determine the mean and standard deviation of the number of frogs with the trait he should expect to find in his sample.
b) Verify that he can use a Normal model to approximate the distribution of the number of frogs with the trait.
c) He found the trait in 22 of his frogs. Do you think this proves that the trait has become more common? Explain.

Sheryl Ezze

Numerade Educator

Problem 43

Lefties, again A lecture hall has 170 seats with folding arm tablets, 27 of which are designed for left-handers. The average size of classes that meet there is 150 , and we can assume that about $12 \%$ of students are left-handed. What's the probability that a right-handed student in one of these classes is forced to use a lefty arm tablet?

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 44

No-shows An airline, believing that $5 \%$ of passengers fail to show up for flights, overbooks (sells more tickets than there are seats). Suppose a plane will hold 265 passengers, and the airline sells 275 tickets. What's the probability the airline will not have enough seats, so someone gets bumped?

Michelle Z.

Numerade Educator

Problem 45

Annoying phone calls A newly hired telemarketer is told he will probably make a sale on about $15 \%$ of his phone calls. The first week he called 150 people, but only made 20 sales. Should he suspect he was misled about the true success rate? Explain.

Nicholas Salas

Numerade Educator

Problem 46

The euro Shortly after the introduction of the euro coin in Belgium, newspapers around the world published articles claiming the coin is biased. The stories were based on reports that someone had spun the coin 250 times and gotten 140 heads - that's $56 \%$ heads. Do you think this is evidence that spinning a euro is unfair? Explain.

Robin Corrigan

Numerade Educator

Problem 47

Hurricanes, redux We first looked at the occurrences of hurricanes in Chapter 3 (Exercise 47 ). Suppose we find and the arrivals can be modeled by a Poisson distribution with mean 2.45 .
a) What's the probability of no hurricanes next year?
b) What's the probability that during the next two years, there's exactly 1 hurricane?

James Kiss

Numerade Educator

Problem 48

Bank tellers I am the only bank teller on duty at my local bank. I need to run out for 10 minutes, but I don't want to miss any customers. Suppose the arrival of customers can be modeled by a Poisson distribution with mean 2 customers per hour.
a) What's the probability that no one will arrive in the next 10 minutes?
b) What's the probability that 2 or more people arrive in the next 10 minutes?
c) You've just served 2 customers who came in one after the other. Is this a better time to run out?

James Kiss

Numerade Educator

Problem 49

TB, again In Chapter 14 we saw that the probability of contracting TB is small, with $p$ about 0.0005 for a new case in a given year. In a town of 6000 people:
a) What's the expected number of new cases?
b) Use the Poisson model to approximate the probability that there will be at least one new case of TB next year.

James Kiss

Numerade Educator

Problem 50

Earthquakes Suppose the probability of a major earthquake on a given day is 1 out of 20,000 .
a) What's the expected number of major earthquakes in the next 2000 days?
b) Use the Poisson model to approximate the probability that there will be at least one major earthquake in the next 2000 days.

James Kiss

Numerade Educator

Problem 51

Seatbelts II Police estimate that $80 \%$ of drivers now wear their seatbelts. They set up a safety roadblock, stopping cars to check for seatbelt use.
a) How many cars do they expect to stop before finding a driver whose seatbelt is not buckled?
b) What's the probability that the first unbelted driver is in the sixth car stopped?
c) What's the probability that the first 10 drivers are all wearing their seatbelts?
d) If they stop 30 cars during the first hour, find the mean and standard deviation of the number of drivers expected to be wearing seatbelts.
e) If they stop 120 cars during this safety check, what's the probability they find at least 20 drivers not wearing their seatbelts?

Sheryl Ezze

Numerade Educator

Problem 52

Rickets Vitamin D is essential for strong, healthy bones. Our bodies produce vitamin D naturally when sunlight falls upon the skin, or it can be taken as a dietary supplement. Although the bone disease rickets was largely eliminated in England during the $1950 \mathrm{~s}$, some people there are concerned that this generation of children is at increased risk because they are more likely to watch TV or play computer games than spend time outdoors. Recent research indicated that about $20 \%$ of British children are deficient in vitamin
D. Suppose doctors test a group of elementary school children.
a) What's the probability that the first vitamin D- deficient child is the eighth one tested?
b) What's the probability that the first 10 children tested are all okay?
c) How many kids do they expect to test before finding one who has this vitamin deficiency?
d) They will test 50 students at the third-grade level. Find the mean and standard deviation of the number who may be deficient in vitamin $\mathrm{D}$.
e) If they test 320 children at this school, what's the probability that no more than 50 of them have the vitamin deficiency?

Sheryl Ezze

Numerade Educator

Problem 53

ESP Scientists wish to test the mind-reading ability of a person who claims to have ESP. They use five cards with different and distinctive symbols (square, circle, triangle, line, squiggle). Someone picks a card at random and thinks about the symbol. The "mind reader" must correctly identify which symbol was on the card. If the test consists of 120 trials, how many would this person need to get right in order to convince you that ESP may actually exist? Explain.

James Kiss

Numerade Educator

Problem 54

True-false A true-false test consists of 50 questions. How many does a student have to get right to convince you that he is not merely guessing? Explain.

Robin Corrigan

Numerade Educator

Problem 55

Hot hand A basketball player who ordinarily makes about $50 \%$ of his free throw shots has made 6 in a row. Is this evidence that he has a "hot hand" tonight? That is, is this streak so unusual that it means the probability he makes a shot must have changed? Explain.

James Kiss

Numerade Educator

Problem 56

New bow The archer in Exercise 30 purchases a new bow, hoping that it will improve her success rate to more than $80 \%$ bull's-eyes. She is delighted when she first tests her new bow and hits 6 consecutive bull's-eyes. Do you think this is compelling evidence that the new bow is better? In other words, is a streak like this unusual for her? Explain.

Ahmad Reda

Numerade Educator

Problem 57

Hotter hand The basketball player in Exercise 55 has new sneakers, which he thinks improve his game. Over his past 60 shots, he's made 32 - much better than the $50 \%$ he usually shoots. Do you think his chances of making a shot really increased? In other words, is making at least 32 of 60 shots really unusual for him? (Do you think it's his sneakers?)

James Kiss

Numerade Educator

Problem 58

New bow, again The archer in Exercise 56 continues shooting arrows, ending up with 45 bull's-eyes in 50 shots. Now are you convinced that the new bow is better? Explain.

James Kiss

Numerade Educator

Problem 59

Web visitors A website manager has noticed that during the evening hours, about 4 people per minute check out from their shopping cart and make an online purchase. She believes that each purchase is independent of the others and wants to model the number of purchases per minute.
a) What model might you suggest to model the number of purchases per minute?
b) What is the probability that in any one minute at least one purchase is made?
c) What is the probability that no one makes a purchase in the next two minutes?

James Kiss

Numerade Educator

Problem 60

Quality control In an effort to improve the quality of their cell phones, a manufacturing manager records the number of faulty phones in each day's production run. The manager notices that the number of faulty cell phones in a production run of cell phones is usually small and that the quality of one day's run seems to have no bearing on the next day.
a) What model might you use to model the number of faulty cell phones produced in one day?
b) If the mean number of faulty cell phones is 3.6 per day, what is the probability that no faulty cell phones will be produced tomorrow?
c) If the mean number of faulty cell phones is 3.6 per day, what is the probability that 3 or more faulty cell phones were produced in today's run?

Rashmi Sinha

Numerade Educator

Problem 61

Web visitors, part 2 The website manager in Exercise 59 wants to model the time between purchases. Recall that the mean number of purchases in the evening is 4 per minute.
a) What model would you use to model the time between events?
b) What is the mean time between purchases?
c) What is the probability that the time to the next purchase will be between 1 and 3 minutes?

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 62

Quality control, part 2 The cell phone manufacturer in Exercise 60 wants to model the time between events. The mean number of defective cell phones is 2 per day.
a) What model would you use to model the time between events?
b) What would the probability be that the time to the next failure is 1 day or less?
c) What is the mean time between failures?

Rashmi Sinha

Numerade Educator