Stats Modeling the World 4th

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

Chapter 15

Random Variables - all with Video Answers

Educators

Chapter Questions

Problem 1

Find the expected value of each random variable:
a) $\begin{array}{l|l|l|l}x & 10 & 20 & 30 \\\hline P(X=x) & 0.3 & 0.5 & 0.2\end{array}$
b) $\begin{array}{l|c|c|c|c}x & 2 & 4 & 6 & 8 \\\hline P(X=x) & 0.3 & 0.4 & 0.2 & 0.1\end{array}$

Kari Hasz

Kari Hasz

Numerade Educator

Problem 2

Find the expected value of each random variable:
a) $\begin{array}{c|c|c|c}x & 0 & 1 & 2 \\\hline P(X=x) & 0.2 & 0.4 & 0.4\end{array}$
b) $\begin{array}{c|c|c|c|c}x & 100 & 200 & 300 & 400 \\\hline P(X=x) & 0.1 & 0.2 & 0.5 & 0.2\end{array}

Arulmozhi T

Numerade Educator

Problem 3

A citrus farmer has observed the following distribution for the number of oranges per tree. How many oranges does he expect on average?
$$\begin{array}{l|c|c|c|c}\text { Oranges } & 25 & 30 & 35 & 40 \\\hline \text { Probability } & 0.10 & 0.40 & 0.30 & 0.20\end{array}$$

Carson Merrill

Numerade Educator

Problem 4

A coffee shop tracks sales and has observed the distribution in the following table. What is the average daily sales that it can expect?
$$\begin{array}{|l|l|l|l|l|l|}\text { # of Sales } & 145 & 150 & 155 & 160 & 170 \\\hline \text { Probability } & 0.15 & 0.22 & 0.37 & 0.19 & 0.07\end{array}$$

Kari Hasz

Numerade Educator

Problem 5

What is the standard deviation for Exercise 3?

Kari Hasz

Numerade Educator

Problem 6

What is the standard deviation for Exercise 4?

Kari Hasz

Numerade Educator

Problem 7

You draw a card from a deck. If you get a red card, you win nothing. If you get a spade, you win $\$ 5 .$ For any club, you win $\$ 10$ plus an extra $\$ 20$ for the ace of clubs.
a) Create a probability model for the amount you win.
b) Find the expected amount you'll win.
c) What would you be willing to pay to play this game?

Kari Hasz

Numerade Educator

Problem 8

You roll a die. If it comes up a $6,$ you win $\$ 100 .$ If not, you get to roll again. If you get a 6 the second time, you win $\$ 50 .$ If not, you lose.
a) Create a probability model for the amount you win.
b) Find the expected amount you'll win.
c) What would you be willing to pay to play this game?

Kari Hasz

Numerade Educator

Problem 9

A couple plans to have children until they get a girl, but they agree that they will not have more than three children even if all are boys. (Assume boys and girls are equally likely.)
a) Create a probability model for the number of children they might have.
b) Find the expected number of children.
c) Find the expected number of boys they’ll have.

Oluwadamilola Ameobi

Oluwadamilola Ameobi

Numerade Educator

Problem 10

A carnival game offers a $\$ 100$ cash prize for anyone who can break a balloon by throwing a dart at it. It costs $\$ 5$ to play, and you're willing to spend up to $\$ 20$ trying to win. You estimate that you have about a $10 \%$ chance of hitting the balloon on any throw.
a) Create a probability model for this carnival game.
b) Find the expected number of darts you'll throw.
c) Find your expected winnings.

Kari Hasz

Numerade Educator

Problem 11

A small software company bids on two contracts. It anticipates a profit of $\$ 60,000$ if it gets the larger contract and a profit of $\$ 20,000$ on the smaller contract. The company estimates there's a $30 \%$ chance it will get the larger contract and a $60 \%$ chance it will get the smaller contract. Assuming the contracts will be awarded independently, what's the expected profit?

Arulmozhi T

Numerade Educator

Problem 12

A man buys a racehorse for $\$ 20,000$ and enters it in two races. He plans to sell the horse afterward, hoping to make a profit. If the horse wins both races, its value will jump to $\$ 100,000 .$ If it wins one of the races, it will be worth $\$ 50,000 .$ If it loses both races, it will be worth only $\$ 10,000 .$ The man believes there's a $20 \%$ chance that the horse will win the first race and a $30 \%$ chance it will win the second one. Assuming that the two races are independent events, find the man's expected profit.

Arulmozhi T

Numerade Educator

Problem 13

Find the standard deviations of the random variables in Exercise 1

Arulmozhi T

Numerade Educator

Problem 14

Find the standard deviations of the random variables in Exercise 2

Arulmozhi T

Numerade Educator

Problem 15

Find the standard deviation of the amount you might win drawing a card in Exercise 7

Kari Hasz

Numerade Educator

Problem 16

Find the standard deviation of the amount you might win rolling a die in Exercise $8 .$

Arulmozhi T

Numerade Educator

Problem 17

Find the standard deviation of the number of children the couple in Exercise 9 may have.

Arulmozhi T

Numerade Educator

Problem 18

Find the standard deviation of your winnings throwing darts in Exercise $10 .$

Arulmozhi T

Numerade Educator

Problem 19

The probability model below describes the number of repair calls that an appliance repair shop may receive during an hour.
$$\begin{array}{l|c|c|c|c}\text { Repair Calls } & 0 & 1 & 2 & 3 \\\hline \text { Probability } & 0.1 & 0.3 & 0.4 & 0.2\end{array}$$
a) How many calls should the shop expect per hour?
b) What is the standard deviation?

Kari Hasz

Numerade Educator

Problem 20

A commuter must pass through five traffic lights on her way to work and will have to stop at each one that is red. She estimates the probability model for the number of red lights she hits, as shown below.
$$\begin{array}{l|c|c|c|c|c|c}\chi=\# \text { of Red } & 0 & 1 & 2 & 3 & 4 & 5 \\\hline P(X=x) & 0.05 & 0.25 & 0.35 & 0.15 & 0.15 & 0.05\end{array}$$
a) How many red lights should she expect to hit each day?
b) What's the standard deviation?

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 21

A consumer organization inspecting new cars found that many had appearance defects (dents, scratches, paint chips, etc.). While none had more than three of these defects, $7 \%$ had three, $11 \%$ two, and $21 \%$ one defect. Find the expected number of appearance defects in a new car and the standard deviation.

Arulmozhi T

Numerade Educator

Problem 22

An insurance policy costs $\$ 100$ and will pay policyholders $\$ 10,000$ if they suffer a major injury (resulting in hospitalization) or $\$ 3000$ if they suffer a minor injury (resulting in lost time from work). The company estimates that each year 1 in every 2000 policyholders may have a major injury, and 1 in 500 a minor injury only.
a) Create a probability model for the profit on a policy.
b) What's the company's expected profit on this policy?
c) What's the standard deviation?

Arulmozhi T

Numerade Educator

Problem 23

Mary is deciding whether to book the cheaper flight home from college after her final exams, but she's unsure when her last exam will be. She thinks there is only a $20 \%$ chance that the exam will be scheduled after the last day she can get a seat on the cheaper flight. If it is and she has to cancel the flight, she will lose $\$ 150 .$ If she can take the cheaper flight, she will save $\$ 100$
a) If she books the cheaper flight, what can she expect to gain, on average?
b) What is the standard deviation?

Kari Hasz

Numerade Educator

Problem 24

An option to buy a stock is priced at $\$ 200$ If the stock closes above 30 on May $15,$ the option will be worth $\$ 1000 .$ If it closes below $20,$ the option will be worth nothing, and if it closes between 20 and 30 (inclusively), the option will be worth $\$ 200 .$ A trader thinks there is a $50 \%$ chance that the stock will close in the $20-30$ range, a $20 \%$ chance that it will close above 30 and a $30 \%$ chance that it will fall below 20 on May 15
a) Should she buy the stock option?
b) How much does she expect to gain?
c) What is the standard deviation of her gain?

Kari Hasz

Numerade Educator

Problem 25

You play two games against the same opponent. The probability you win the first game is 0.4. If you win the first game, the probability you also win the second is $0.2 .$ If you lose the first game, the probability that you win the second is 0.3
a) Are the two games independent? Explain.
b) What's the probability you lose both games?
c) What's the probability you win both games?
d) Let random variable $X$ be the number of games you win. Find the probability model for $X .$
e) What are the expected value and standard deviation?

Bryan Meares

Numerade Educator

Problem 26

Your company bids for two contracts. You believe the probability you get contract #1 is 0.8. If you get contract $\# 1,$ the probability you also get contract $\# 2$ will be $0.2,$ and if you do not get $\# 1,$ the probability you get #2 will be 0.3.
a) Are the two contracts independent? Explain.
b) Find the probability you get both contracts.
c) Find the probability you get no contract.
d) Let $X$ be the number of contracts you get. Find the probability model for $X$
e) Find the expected value and standard deviation.

Arulmozhi T

Numerade Educator

Problem 27

In a group of 10 batteries, 3 are dead. You choose 2 batteries at random.
a) Create a probability model for the number of good batteries you get.
b) What's the expected number of good ones you get?
c) What's the standard deviation?

Arulmozhi T

Numerade Educator

Problem 28

In a litter of seven kittens, three are female. You pick two kittens at random.
a) Create a probability model for the number of male kittens you get.
b) What's the expected number of males?
c) What's the standard deviation?

Kari Hasz

Numerade Educator

Problem 29

Given independent random variables with means and standard deviations as shown, find the mean and standard deviation of:
$$\begin{array}{cc|c} & \text { Mean } & \text { SD } \\\hline X & 10 & 2 \\Y & 20 & 5\end{array}$$
a) $3 X$
b) $Y+6$
c) $X+Y$
d) $X-Y$
e) $X_{1}+X_{2}$

Kari Hasz

Numerade Educator

Problem 30

Given independent random variables with means and standard deviations as shown, find the mean and standard deviation of:
$$\begin{array}{c|c|c} & \text { Mean } & \text { SD } \\\hline x & 80 & 12 \\Y & 12 & 3\end{array}$$
a) $X-20$
b) $0.5 Y$
c) $X+Y$
d) $X-Y$
e) $Y_{1}+Y_{2}$

Kari Hasz

Numerade Educator

Problem 31

Given independent random variables with means and standard deviations as shown, find the mean and standard deviation of:
$$\begin{array}{c|c|c} & \text { Mean } & \text { SD } \\\hline \boldsymbol{X} & 120 & 12 \\\boldsymbol{Y} & 300 & 16\end{array}$$
a) $0.8 Y$
b) $2 X-100$
c) $X+2 Y$
d) $3 X-Y$
e) $Y_{1}+Y_{2}$

Kari Hasz

Numerade Educator

Problem 32

Given independent random variables with means and standard deviations as shown, find the mean and standard deviation of:
$$\begin{array}{c|c|c} & \text { Mean } & \text { SD } \\\hline X & 80 & 12 \\Y & 12 & 3\end{array}$$
a) $2 Y+20$
b) $3 X$
c) $0.25 X+Y$
d) $X-5 Y$
e) $X_{1}+X_{2}+X_{3}$

Kari Hasz

Numerade Educator

Problem 33

An employer pays a mean salary for a 5 -day workweek of $\$ 1250$ with a standard deviation of $\$ 129$ On the weekends, his salary expenses have a mean of \$450 with a standard deviation of $\$ 57 .$ What is the mean and standard deviation of his total weekly salaries?

Kari Hasz

Numerade Educator

Problem 34

A golfer keeps track of his score for playing nine holes of golf (half a normal golf round). His mean score is 85 with a standard deviation of $11 .$ Assuming that the second 9 has the same mean and standard deviation, what is the mean and standard deviation of his total score if he plays a full 18 holes?

Kari Hasz

Numerade Educator

Problem 35

A grocery supplier believes that in a dozen eggs, the mean number of broken ones is 0.6 with a standard deviation of 0.5 eggs. You buy 3 dozen eggs without checking them.
a) How many broken eggs do you expect to get?
b) What's the standard deviation?
c) What assumptions did you have to make about the eggs in order to answer this question?

Kari Hasz

Numerade Educator

Problem 36

A company selling vegetable seeds in packets of 20 estimates that the mean number of seeds that will actually grow is $18,$ with a standard deviation of 1.2 seeds. You buy 5 different seed packets.
a) How many good seeds do you expect to get?
b) What's the standard deviation?
c) What assumptions did you make about the seeds? Do you think that assumption is warranted? Explain.

Prashant Bana

Numerade Educator

Problem 37

In Exercise 35 you bought 3 dozen eggs.
a) How many good eggs do you expect?
b) What's the standard deviation of the number of good eggs?
c) Why does it make sense for the standard deviation not to change?

Arulmozhi T

Numerade Educator

Problem 38

In Exercise 36 you bought 5 seed packets.
a) How many bad seeds do you expect?
b) What is the standard deviation of the bad seed count?
c) Why does it make sense that the standard deviation is not different?

Arulmozhi T

Numerade Educator

Problem 39

Remember back in Chapter 5 when we used the equation $S A T=40 \times A C T+150$ to convert an ACT score into a SAT score. Let's use this transformation again, now with random variables.
a) Suppose your school has a mean ACT score of 29 What would its equivalent mean SAT score be?
b) If your school has a standard deviation of $5 \mathrm{ACT}$ points, what is the standard of its equivalent SAT score?

Arulmozhi T

Numerade Educator

Problem 40

We used the formula $^{\circ} F=9 / 5^{\circ} \mathrm{C}+32$ to convert Celsius to Fahrenheit in Chapter $5 .$ Let's put it to use in a random variable setting.
a) Suppose your town has a mean January temperature of $11^{\circ} \mathrm{C} .$ What is the mean temperature in $^{\circ} \mathrm{F} ?$
b) Fortunately your local weatherman has recently taken a statistics course and is keen to show off his newfound knowledge. He reports that January has a standard deviation of $6^{\circ} \mathrm{C} .$ What is the standard deviation in $^{\circ} \mathrm{F} ?$

Arulmozhi T

Numerade Educator

Problem 41

Suppose that the appliance shop in Exercise 19 plans an 8 -hour day.
a) Find the mean and standard deviation of the number of repair calls they should expect in a day.
b) What assumption did you make about the repair calls?
c) Use the mean and standard deviation to describe what a typical 8-hour day will be like.
d) At the end of a day, a worker comments "Boy, I'm tired. Today was sure unusually busy!" How many repair calls would justify such an observation?

Arulmozhi T

Numerade Educator

Problem 42

Suppose the commuter in Exercise 20 has a 5 -day workweek.
a) Find the mean and standard deviation of the number of red lights the commuter should expect to hit in her week.
b) What assumption did you make about the days?
c) Use the mean and standard deviation to describe a typical week.
d) Upon arriving home on Friday, the commuter remarks, "Wow! My commute was quick all week." How many red lights would it take to deserve a feeling of good luck?

Kari Hasz

Numerade Educator

Problem 43

A delivery company's trucks occasionally get parking tickets, and based on past experience, the company plans that the trucks will average 1.3 tickets a month, with a standard deviation of 0.7 tickets.
a) If they have 18 trucks, what are the mean and standard deviation of the total number of parking tickets the company will have to pay this month?
b) What assumption did you make in answering?

Arulmozhi T

Numerade Educator

Problem 44

Organizers of a televised fundraiser know from past experience that most people donate small amounts $(\$ 10-\$ 25),$ some donate larger amounts $(\$ 50-$ $\$ 100$ ), and a few people make very generous donations of $\$ 250, \$ 500,$ or more. Historically, pledges average about $\$ 32$ with a standard deviation of $\$ 54$
a) If 120 people call in pledges, what are the mean and standard deviation of the total amount raised?
b) What assumption did you make in answering this question?

Arulmozhi T

Numerade Educator

Problem 45

An insurance company estimates that it should make an annual profit of $\$ 150$ on each homeowner's policy written, with a standard deviation of $\$ 6000$
a) Why is the standard deviation so large?
b) If it writes only two of these policies, what are the mean and standard deviation of the annual profit?
c) If it writes 10,000 of these policies, what are the mean and standard deviation of the annual profit?
d) Is the company likely to be profitable? Explain.
e) What assumptions underlie your analysis? Can you think of circumstances under which those assumptions might be violated? Explain.

Kari Hasz

Numerade Educator

Problem 46

A casino knows that people play the slot machines in hopes of hitting the jackpot but that most of them lose their dollar. Suppose a certain machine pays out an average of $\$ 0.92,$ with a standard deviation of $\$ 120.$
a) Why is the standard deviation so large?
b) If you play 5 times, what are the mean and standard deviation of the casino's profit?
c) If gamblers play this machine 1000 times in a day, what are the mean and standard deviation of the casino's profit?
d) Is the casino likely to be profitable? Explain.

Kari Hasz

Numerade Educator

Problem 47

The amount of cereal that can be poured into a small bowl varies with a mean of 1.5 ounces and a standard deviation of 0.3 ounces. A large bowl holds a mean of 2.5 ounces with a standard deviation of 0.4 ounces. You open a new box of cereal and pour one large and one small bowl.
a) How much more cereal do you expect to be in the large bowl?
b) What's the standard deviation of this difference?
c) If the difference follows a Normal model, what's the probability the small bowl contains more cereal than the large one?
d) What are the mean and standard deviation of the total amount of cereal in the two bowls?
e) If the total follows a Normal model, what's the probability you poured out more than 4.5 ounces of cereal in the two bowls together?
f) The amount of cereal the manufacturer puts in the boxes is a random variable with a mean of 16.3 ounces and a standard deviation of 0.2 ounces. Find the expected amount of cereal left in the box and the standard deviation.

Kari Hasz

Numerade Educator

Problem 48

The American Veterinary Association claims that the annual cost of medical care for dogs averages $\$ 100,$ with a standard deviation of $\$ 30,$ and for cats averages $\$ 120$ with a standard deviation of $\$ 35$
a) What's the expected difference in the cost of medical care for dogs and cats?
b) What's the standard deviation of that difference?
c) If the costs can be described by Normal models, what's the probability that medical expenses are higher for someone's dog than for her cat?
d) What concerns do you have?

Kari Hasz

Numerade Educator

Problem 49

In Exercise 47 we poured a large and a small bowl of cereal from a box. Suppose the amount of cereal that the manufacturer puts in the boxes is a random variable with mean 16.2 ounces and standard deviation 0.1 ounces.
a) Find the expected amount of cereal left in the box.
b) What's the standard deviation?
c) If the weight of the remaining cereal can be described by a Normal model, what's the probability that the box still contains more than 13 ounces?

Kari Hasz

Numerade Educator

Problem 50

You're thinking about getting two dogs and a cat. Assume that annual veterinary expenses are independent and have a Normal model with the means and standard deviations described in Exercise $48 .$
a) Define appropriate variables and express the total annual veterinary costs you may have.
b) Describe the model for this total cost. Be sure to specify its name, expected value, and standard deviation.
c) What's the probability that your total expenses will exceed $\$ 400 ?$

Kari Hasz

Numerade Educator

Problem 51

In the $4 \times 100$ medley relay event, four swimmers swim 100 yards, each using a different stroke. $A$ college team preparing for the conference championship looks at the times their swimmers have posted and creates a model based on the following assumptions:
$\bullet$ The swimmers' performances are independent.
$\bullet$ Each swimmer's times follow a Normal model.
$\bullet$ The means and standard deviations of the times (in seconds) are as shown:
$$\begin{array}{l|l|c}\text { Swimmer } & \text { Mean } & \text { SD } \\\hline 1 \text { (backstroke) } & 50.72 & 0.24 \\2 \text { (breaststroke) } & 55.51 & 0.22 \\3 \text { (butterfy) } & 49.43 & 0.25 \\4 \text { (freestyle) } & 44.91 & 0.21\end{array}$$
a) What are the mean and standard deviation for the relay team's total time in this event?
b) The team's best time so far this season was 3: 19.48 (That's 199.48 seconds.) Do you think the team is likely to swim faster than this at the conference championship? Explain.

Bryan Meares

Numerade Educator

Problem 52

Bicycles arrive at a bike shop in boxes. Before they can be sold, they must be unpacked, assembled, and tuned (lubricated, adjusted, etc.). Based on past experience, the shop manager makes the following assumptions about how long this may take:
$\bullet$ The times for each setup phase are independent.
$\bullet$ The times for each phase follow a Normal model.
$\bullet$ The means and standard deviations of the times (in minutes) are as shown:
$$\begin{array}{l|c|c}\text { Phase } & \text { Mean } & \text { SD } \\\hline \text { Unpacking } & 3.5 & 0.7 \\\text { Assembly } & 21.8 & 2.4 \\\text { Tuning } & 12.3 & 2.7\end{array}$$
a) What are the mean and standard deviation for the total bicycle setup time?
b) A customer decides to buy a bike like one of the display models but wants a different color. The shop has one, still in the box. The manager says they can have it ready in half an hour. Do you think the bike will be set up and ready to go as promised? Explain.

Trinity Steen

Numerade Educator

Problem 53

A farmer has 100 lb of apples and 50 lb of potatoes for sale. The market price for apples (per pound) each day is a random variable with a mean of 0.5 dollars and a standard deviation of 0.2 dollars. Similarly, for a pound of potatoes, the mean price is 0.3 dollars and the standard deviation is 0.1 dollars. It also costs him 2 dollars to bring all the apples and potatoes to the market. The market is busy with eager shoppers, so we can assume that he'll be able to sell all of each type of produce at that day's price.
a) Define your random variables, and use them to express the farmer's net income.
b) Find the mean.
c) Find the standard deviation of the net income.
d) Do you need to make any assumptions in calculating the mean? How about the standard deviation?

Kari Hasz

Numerade Educator

Problem 54

The bicycle shop in Exercise 52 will be offering 2 specially priced children's models at a sidewalk sale. The basic model will sell for $\$ 120$ and the deluxe model for $\$ 150 .$ Past experience indicates that sales of the basic model will have a mean of 5.4 bikes with a standard deviation of $1.2,$ and sales of the deluxe model will have a mean of 3.2 bikes with a standard deviation of 0.8 bikes. The cost of setting up for the sidewalk sale is $\$ 200.$
a) Define random variables and use them to express the bicycle shop's net income.
b) What's the mean of the net income?
c) What's the standard deviation of the net income?
d) Do you need to make any assumptions in calculating the mean? How about the standard deviation?

Kari Hasz

Numerade Educator

Problem 55

At a certain coffee shop, all the customers buy a cup of coffee; some also buy a doughnut. The shop owner believes that the number of cups he sells each day is normally distributed with a mean of 320 cups and a standard deviation of 20 cups. He also believes that the number of doughnuts he sells each day is independent of the coffee sales and is normally distributed with a mean of 150 doughnuts and a standard deviation of 12.
a) The shop is open every day but Sunday. Assuming day-to-day sales are independent, what's the probability he'll sell over 2000 cups of coffee in a week?
b) If he makes a profit of 50 cents on each cup of coffee and 40 cents on each doughnut, can he reasonably expect to have a day's profit of over $\$ 300 ?$ Explain.
c) What's the probability that on any given day he'll sell a doughnut to more than half of his coffee customers?

Kari Hasz

Numerade Educator

Problem 56

The Atlas BodyBuilding Company (ABC) sells "starter sets" of barbells that consist of one bar, two 20-pound weights, and four 5 -pound weights. The bars weigh an average of 10 pounds with a standard deviation of 0.25 pounds. The weights average the specified amounts, but the standard deviations are 0.2 pounds for the 20 -pounders and 0.1 pounds for the 5 -pounders. We can assume that all the weights are normally distributed.
a) ABC ships these starter sets to customers in two boxes: The bar goes in one box and the six weights go in another. What's the probability that the total weight in that second box exceeds 60.5 pounds? Define your variables clearly and state any assumptions you make.
b) It costs ABC $\$ 0.40$ per pound to ship the box containing the weights. Because it's an odd-shaped package, though, shipping the bar costs $\$ 0.50$ a pound plus a $\$ 6.00$ surcharge. Find the mean and standard deviation of the company's total cost for shipping a starter set.
c) Suppose a customer puts a 20 -pound weight at one end of the bar and the four 5 -pound weights at the other end. Although he expects the two ends to weigh the same, they might differ slightly. What's the probability the difference is more than a quarter of a pound?

Kari Hasz

Numerade Educator