Statistics for Business Economics

David R. Anderson, Dennis J. Sweeney, Thomas A. Williams

Chapter 4

Introduction to Probability - all with Video Answers

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Chapter Questions

Problem 1

An experiment has three steps with three outcomes possible for the first step, two outcomes possible for the second step, and four outcomes possible for the third step. How many experimental outcomes exist for the entire experiment?

Joshua Hale

Joshua Hale

Numerade Educator

Problem 2

How many ways can three items be selected from a group of six items? Use the letters $A, B$ $\mathrm{C}, \mathrm{D}, \mathrm{E},$ and $\mathrm{F}$ to identify the items, and list each of the different combinations of three items.

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 3

How many permutations of three items can be selected from a group of six? Use the letters $A$, $\mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E},$ and $\mathrm{F}$ to identify the items, and list each of the permutations of items $\mathrm{B}, \mathrm{D},$ and $\mathrm{F}$.

Sheryl Ezze

Numerade Educator

Problem 4

Consider the experiment of tossing a coin three times.
a. Develop a tree diagram for the experiment.
b. List the experimental outcomes.
c. What is the probability for each experimental outcome?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 5

Suppose an experiment has five equally likely outcomes: $E_{1}, E_{2}, E_{3}, E_{4}, E_{5} .$ Assign probabilities to each outcome and show that the requirements in equations (4.3) and (4.4) are satisfied. What method did you use?

Jon Southam

Numerade Educator

Problem 6

An experiment with three outcomes has been repeated 50 times, and it was learned that $E_{1}$ occurred 20 times, $E_{2}$ occurred 13 times, and $E_{3}$ occurred 17 times. Assign probabilities to the outcomes. What method did you use?

SY

Song Yh

Numerade Educator

Problem 7

A decision maker subjectively assigned the following probabilities to the four outcomes of an experiment: $P\left(E_{1}\right)=.10, P\left(E_{2}\right)=.15, P\left(E_{3}\right)=.40,$ and $P\left(E_{4}\right)=.20 .$ Are these probability assignments valid? Explain.

Joshua Hale

Numerade Educator

Problem 8

In the city of Milford, applications for zoning changes go through a two-step process: a review by the planning commission and a final decision by the city council. At step 1 the planning commission reviews the zoning change request and makes a positive or negative recommendation concerning the change. At step 2 the city council reviews the planning commission's recommendation and then votes to approve or to disapprove the zoning change. Suppose the developer of an apartment complex submits an application for a zoning change. Consider the application process as an experiment.
a. How many sample points are there for this experiment? List the sample points.
b. Construct a tree diagram for the experiment.

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 9

Simple random sampling uses a sample of size $n$ from a population of size $N$ to obtain data that can be used to make inferences about the characteristics of a population. Suppose that, from a population of 50 bank accounts, we want to take a random sample of four accounts in order to learn about the population. How many different random samples of four accounts are possible?

Sheryl Ezze

Numerade Educator

Problem 10

Many students accumulate debt by the time they graduate from college. Shown in the following table is the percentage of graduates with debt and the average amount of debt for these graduates at four universities and four liberal arts colleges (U.S. News and World Report, America's Best Colleges, 2008 ).
a. If you randomly choose a graduate of Morehouse College, what is the probability that this individual graduated with debt?
b. If you randomly choose one of these eight institutions for a follow-up study on student loans, what is the probability that you will choose an institution with more than $60 \%$ of its graduates having debt?
c. If you randomly choose one of these eight institutions for a follow-up study on student loans, what is the probability that you will choose an institution whose graduates with debts have an average debt of more than $\$ 30,000 ?$
d. What is the probability that a graduate of Pace University does not have debt?
e. For graduates of Pace University with debt, the average amount of debt is $\$ 32,980$ Considering all graduates from Pace University, what is the average debt per graduate?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 11

The National Highway Traffic Safety Administration (NHTSA) conducted a survey to learn about how drivers throughout the United States are using seat belts (Associated Press, August 25,2003 ). Sample data consistent with the NHTSA survey are as follows.
a. For the United States, what is the probability that a driver is using a seat belt?
b. The seat belt usage probability for a U.S. driver a year earlier was .75. NHTSA chief Dr. Jeffrey Runge had hoped for a .78 probability in $2003 .$ Would he have been pleased with the 2003 survey results?
c. What is the probability of seat belt usage by region of the country? What region has the highest seat belt usage?
d. What proportion of the drivers in the sample came from each region of the country? What region had the most drivers selected? What region had the second most drivers selected?
e. Assuming the total number of drivers in each region is the same, do you see any reason why the probability estimate in part (a) might be too high? Explain.

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 12

The Powerball lottery is played twice each week in 28 states, the Virgin Islands, and the District of Columbia. To play Powerball a participant must purchase a ticket and then select five numbers from the digits 1 through 55 and a Powerball number from the digits 1 through
42. To determine the winning numbers for each game, lottery officials draw five white balls out of a drum with 55 white balls, and one red ball out of a drum with 42 red balls. To win the jackpot, a participant's numbers must match the numbers on the five white balls in any order and the number on the red Powerball. Eight coworkers at the ConAgra Foods plant in Lincoln, Nebraska, claimed the record $\$ 365$ million jackpot on February $18,2006,$ by matching the numbers $15-17-43-44-49$ and the Powerball number $29 .$ A variety of other cash prizes are awarded each time the game is played. For instance, a prize of $\$ 200,000$ is paid if the participant's five numbers match the numbers on the five white balls (Powerball website, March 19,2006 ).
a. Compute the number of ways the first five numbers can be selected.
b. What is the probability of winning a prize of $\$ 200,000$ by matching the numbers on the five white balls?
c. What is the probability of winning the Powerball jackpot?

Audrey Fong

Numerade Educator

Problem 13

A company that manufactures toothpaste is studying five different package designs. Assuming that one design is just as likely to be selected by a consumer as any other design, what selection probability would you assign to each of the package designs? In an actual experiment, 100 consumers were asked to pick the design they preferred. The following data were obtained. Do the data confirm the belief that one design is just as likely to be selected as another? Explain.

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 14

An experiment has four equally likely outcomes: $E_{1}, E_{2}, E_{3},$ and $E_{4}$
a. What is the probability that $E_{2}$ occurs?
b. What is the probability that any two of the outcomes occur (e.g., $E_{1}$ or $E_{3}$ )?
c. What is the probability that any three of the outcomes occur (e.g., $E_{1}$ or $E_{2}$ or $E_{4}$ )?

Manisha Sarker

Numerade Educator

Problem 15

Consider the experiment of selecting a playing card from a deck of 52 playing cards. Each card corresponds to a sample point with a $1 / 52$ probability.
a. List the sample points in the event an ace is selected.
b. List the sample points in the event a club is selected.
c. List the sample points in the event a face card (jack, queen, or king) is selected.
d. Find the probabilities associated with each of the events in parts (a), (b), and

Joshua Hale

Numerade Educator

Problem 16

Consider the experiment of rolling a pair of dice. Suppose that we are interested in the sum of the face values showing on the dice.
a. How many sample points are possible? (Hint: Use the counting rule for multiple-step experiments.
b. List the sample points.
c. What is the probability of obtaining a value of $7 ?$
d. What is the probability of obtaining a value of 9 or greater?
e. Because each roll has six possible even values $(2,4,6,8,10, \text { and } 12)$ and only five possible odd values $(3,5,7,9, \text { and } 11),$ the dice should show even values more often than odd values. Do you agree with this statement? Explain.
f. What method did you use to assign the probabilities requested?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 17

Refer to the KP\&L sample points and sample point probabilities in Tables 4.2 and 4.3
a. The design stage (stage 1) will run over budget if it takes 4 months to complete. List the sample points in the event the design stage is over budget.
b. What is the probability that the design stage is over budget?
c. The construction stage (stage 2) will run over budget if it takes 8 months to complete. List the sample points in the event the construction stage is over budget.
d. What is the probability that the construction stage is over budget?
e. What is the probability that both stages are over budget?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 18

To investigate how often families eat at home, Harris Interactive surveyed 496 adults living with children under the age of $18(U S A \text { Today, January } 3,2007$ ). The survey results are shown in the following table.
For a randomly selected family with children under the age of $18,$ compute the following.
a. The probability the family eats no meals at home during the week.
b. The probability the family eats at least four meals at home during the week.
c. The probability the family eats two or fewer meals at home during the week.

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 19

The National Sporting Goods Association conducted a survey of persons 7 years of age or older about participation in sports activities (Statistical Abstract of the United States, 2002 ). The total population in this age group was reported at 248.5 million, with 120.9 million male and 127.6 million female. The number of participants for the top five sports activities appears here.
a. For a randomly selected female, estimate the probability of participation in each of the sports activities.
b. For a randomly selected male, estimate the probability of participation in each of the sports activities.
c. For a randomly selected person, what is the probability the person participates in exercise walking?
d. Suppose you just happen to see an exercise walker going by. What is the probability the walker is a woman? What is the probability the walker is a man?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 20

Fortune magazine publishes an annual list of the 500 largest companies in the United States. The following data show the five states with the largest number of Fortune 500 companies (The New York Times Almanac, 2006).
Suppose a Fortune 500 company is chosen for a follow-up questionnaire. What are the probabilities of the following events?
a. Let $N$ be the event the company is headquartered in New York. Find $P(N)$
b. Let $T$ be the event the company is headquartered in Texas. Find $P(T)$
c. Let $B$ be the event the company is headquartered in one of these five states. Find $P(B)$

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 21

The U.S. adult population by age is as follows (The World Almanac, 2009). The data are in millions of people.
Assume that a person will be randomly chosen from this population.
a. What is the probability the person is 18 to 24 years old?
b. What is the probability the person is 18 to 34 years old?
c. What is the probability the person is 45 or older?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 22

Suppose that we have a sample space with five equally likely experimental outcomes: $E_{1}$ $E_{2}, E_{3}, E_{4}, E_{5} .$ Let
$$\begin{array}{l}
A=\left\{E_{1}, E_{2}\right\} \\
B=\left\{E_{3}, E_{4}\right\} \\
C=\left\{E_{2}, E_{3}, E_{5}\right\}
\end{array}$$
a. Find $P(A), P(B),$ and $P(C)$
b. Find $P(A \cup B)$. Are $A$ and $B$ mutually exclusive?
c. Find $A^{c}, C^{c}, P\left(A^{c}\right),$ and $P\left(C^{c}\right)$
d. Find $A \cup B^{c}$ and $P\left(A \cup B^{c}\right)$
e. Find $P(B \cup C)$

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 23

Suppose that we have a sample space $S=\left\{E_{1}, E_{2}, E_{3}, E_{4}, E_{5}, E_{6}, E_{7}\right\},$ where $E_{1}, E_{2}, \ldots,$ $E_{7}$ denote the sample points. The following probability assignments apply: $P\left(E_{1}\right)=.05$ $P\left(E_{2}\right)=.20, P\left(E_{3}\right)=.20, P\left(E_{4}\right)=.25, P\left(E_{5}\right)=.15, P\left(E_{6}\right)=.10,$ and $P\left(E_{7}\right)=.05 .$ Let
$$\begin{array}{l}
A=\left\{E_{1}, E_{4}, E_{6}\right\} \\
B=\left\{E_{2}, E_{4}, E_{7}\right\} \\
C=\left\{E_{2}, E_{3}, E_{5}, E_{7}\right\}
\end{array}$$
a. Find $P(A), P(B),$ and $P(C)$
b. Find $A \cup B$ and $P(A \cup B)$
c. Find $A \cap B$ and $P(A \cap B)$
d. Are events $A$ and $C$ mutually exclusive?
e. Find $B^{c}$ and $P\left(B^{c}\right)$

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 24

Clarkson University surveyed alumni to learn more about what they think of Clarkson. One part of the survey asked respondents to indicate whether their overall experience at Clarkson fell short of expectations, met expectations, or surpassed expectations. The results showed that $4 \%$ of the respondents did not provide a response, $26 \%$ said that their experience fell short of expectations, and $65 \%$ of the respondents said that their experience met expectations.
a. If we chose an alumnus at random, what is the probability that the alumnus would say their experience surpassed expectations?
b. If we chose an alumnus at random, what is the probability that the alumnus would say their experience met or surpassed expectations?

Joshua Hale

Numerade Educator

Problem 25

The U.S. Census Bureau provides data on the number of young adults, ages $18-24,$ who are living in their parents' home. $^{1}$ Let
$M=$ the event a male young adult is living in his parents' home $F=$ the event a female young adult is living in her parents' home
If we randomly select a male young adult and a female young adult, the Census Bureau data enable us to conclude $P(M)=.56$ and $P(F)=.42$ (The World Almanac, 2006 ). The probability that both are living in their parents' home is .24
a. What is the probability at least one of the two young adults selected is living in his or her parents' home?
b. What is the probability both young adults selected are living on their own (neither is living in their parents' home)?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 26

Information about mutual funds provided by Morningstar Investment Research includes the type of mutual fund (Domestic Equity, International Equity, or Fixed Income) and the Morningstar rating for the fund. The rating is expressed from 1-star (lowest rating) to 5-star (highest rating). A sample of 25 mutual funds was selected from Morningstar Funds500 (2008) . The following counts were obtained:
$\bullet$ Sixteen mutual funds were Domestic Equity funds.
$\bullet$ Thirteen mutual funds were rated 3 -star or less.
$\bullet$ Seven of the Domestic Equity funds were rated 4-star.
$\bullet$ Two of the Domestic Equity funds were rated 5 -star.
Assume that one of these 25 mutual funds will be randomly selected in order to learn more about the mutual fund and its investment strategy.
a. What is the probability of selecting a Domestic Equity fund?
b. What is the probability of selecting a fund with a 4 -star or 5 -star rating?
c. What is the probability of selecting a fund that is both a Domestic Equity fund and a fund with a 4 -star or 5 -star rating?
d. What is the probability of selecting a fund that is a Domestic Equity fund or a fund with a 4 -star or 5 -star rating?

Joshua Hale

Numerade Educator

Problem 27

What NCAA college basketball conferences have the higher probability of having a team play in college basketball's national championship game? Over the last 20 years, the Atlantic Coast Conference (ACC) ranks first by having a team in the championship game 10 times. The Southeastern Conference (SEC) ranks second by having a team in the championship game 8 times. However, these two conferences have both had teams in the championship game only one time, when Arkansas (SEC) beat Duke (ACC) $76-70$ in 1994 (NCAA website, April 2009 ). Use these data to estimate the following probabilities.
a. What is the probability the ACC will have a team in the championship game?
b. What is the probability the SEC will have team in the championship game?
c. What is the probability the ACC and SEC will both have teams in the championship game?
d. What is the probability at least one team from these two conferences will be in the championship game? That is, what is the probability a team from the ACC or SEC will play in the championship game?
e. What is the probability that the championship game will not a have team from one of these two conferences?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 28

A survey of magazine subscribers showed that $45.8 \%$ rented a car during the past 12 months for business reasons, $54 \%$ rented a car during the past 12 months for personal reasons, and
$30 \%$ rented a car during the past 12 months for both business and personal reasons.
a. What is the probability that a subscriber rented a car during the past 12 months for business or personal reasons?
b. What is the probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons?

Joshua Hale

Numerade Educator

Problem 29

High school seniors with strong academic records apply to the nation's most selective colleges in greater numbers each year. Because the number of slots remains relatively stable, some colleges reject more early applicants. The University of Pennsylvania received 2851 applications for early admission. Of this group, it admitted 1033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration. In the past, Penn has admitted $18 \%$ of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was 2375 (USA Today, January 24,2001 ). Let $E, R,$ and $D$ represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool.
a. Use the data to estimate $P(E), P(R),$ and $P(D)$
b. Are events $E$ and $D$ mutually exclusive? Find $P(E \cap D)$.
c. For the 2375 students admitted to Penn, what is the probability that a randomly selected student was accepted during early admission?
d. Suppose a student applies to Penn for early admission. What is the probability the student will be admitted for early admission or be deferred and later admitted during the regular admission process?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 30

Suppose that we have two events, $A$ and $B,$ with $P(A)=.50, P(B)=.60,$ and $P(A \cap B)=.40$
a. Find $P(A | B)$
b. Find $P(B | A)$
c. Are $A$ and $B$ independent? Why or why not?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 31

Assume that we have two events, $A$ and $B$, that are mutually exclusive. Assume further that we know $P(A)=.30$ and $P(B)=.40$
a. What is $P(A \cap B) ?$
b. What is $P(A | B) ?$
c. $\quad$ A student in statistics argues that the concepts of mutually exclusive events and independent events are really the same, and that if events are mutually exclusive they must be independent. Do you agree with this statement? Use the probability information in this problem to justify your answer.
d. What general conclusion would you make about mutually exclusive and independent events given the results of this problem?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 32

The automobile industry sold 657,000 vehicles in the United States during January 2009 (The Wall Street Journal, February 4,2009 ). This volume was down $37 \%$ from January 2008 as economic conditions continued to decline. The Big Three U.S. automakersGeneral Motors, Ford, and Chrysler-sold 280,500 vehicles, down $48 \%$ from January $2008 .$ A summary of sales by automobile manufacturer and type of vehicle sold is shown in the following table. Data are in thousands of vehicles. The non-U.S. manufacturers are led by Toyota, Honda, and Nissan. The category Light Truck includes pickup, minivan, SUV, and crossover models.
a. Develop a joint probability table for these data and use the table to answer the remaining questions.
b. What are the marginal probabilities? What do they tell you about the probabilities associated with the manufacturer and the type of vehicle sold?
c. If a vehicle was manufactured by one of the U.S. automakers, what is the probability that the vehicle was a car? What is the probability it was a light truck?
d. If a vehicle was not manufactured by one of the U.S. automakers, what is the probability that the vehicle was a car? What is the probability it was a light truck?
e. If the vehicle was a light truck, what is the probability that it was manufactured by one of the U.S. automakers?
f. What does the probability information tell you about sales?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 33

In a survey of MBA students, the following data were obtained on "students' first reason for application to the school in which they matriculated."
a. Develop a joint probability table for these data.
b. Use the marginal probabilities of school quality, school cost or convenience, and other to comment on the most important reason for choosing a school.
c. If a student goes full time, what is the probability that school quality is the first reason for choosing a school?
d. If a student goes part time, what is the probability that school quality is the first reason for choosing a school?
e. Let $A$ denote the event that a student is full time and let $B$ denote the event that the student lists school quality as the first reason for applying. Are events $A$ and $B$ independent? Justify your answer

Rowan Ahmed

Numerade Educator

Problem 34

The U.S. Department of Transportation reported that during November, $83.4 \%$ of Southwest Airlines' flights, $75.1 \%$ of US Airways' flights, and $70.1 \%$ of JetBlue's flights arrived on time (USA Today, January 4, 2007). Assume that this on-time performance is applicable for flights arriving at concourse A of the Rochester International Airport, and that $40 \%$ of the arrivals at concourse A are Southwest Airlines flights, $35 \%$ are US Airways flights, and $25 \%$ are JetBlue flights.
a. Develop a joint probability table with three rows (airlines) and two columns (on-time arrivals vs. late arrivals).
b. $\quad$ An announcement has just been made that Flight 1424 will be arriving at gate 20 in concourse A. What is the most likely airline for this arrival?
c. What is the probability that Flight 1424 will arrive on time?
d. Suppose that an announcement is made saying that Flight 1424 will be arriving late. What is the most likely airline for this arrival? What is the least likely airline?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 35

According to the Ameriprise Financial Money Across Generations study, 9 out of 10 parents with adult children ages 20 to 35 have helped their adult children with some type of financial assistance ranging from college, a car, rent, utilities, credit-card debt, and/or down payments for houses (Money, January 2009 ). The following table with sample data consistent with the study shows the number of times parents have given their adult children financial assistance to buy a car and to pay rent.
a. Develop a joint probability table and use it to answer the remaining questions.
b. Using the marginal probabilities for buy a car and pay rent, are parents more likely to assist their adult children with buying a car or paying rent? What is your interpretation of the marginal probabilities?
c. If parents provided financial assistance to buy a car, what it the probability that the parents assisted with paying rent?
d. If parents did not provide financial assistance to buy a car, what is the probability the parents assisted with paying rent?
e. Is financial assistance to buy a car independent of financial assistance to pay rent? Use probabilities to justify your answer.
f. What is the probability that parents provided financial assistance for their adult children by either helping buy a car or pay rent?

Bailey Brooks

Numerade Educator

Problem 36

Jerry Stackhouse of the National Basketball Association's Dallas Mavericks is the best free-throw shooter on the team, making $89 \%$ of his shots (ESPN website, July, 2008 ). Assume that late in a basketball game, Jerry Stackhouse is fouled and is awarded two shots.
a. What is the probability that he will make both shots?
b. What is the probability that he will make at least one shot?
c. What is the probability that he will miss both shots?
d. Late in a basketball game, a team often intentionally fouls an opposing player in order to stop the game clock. The usual strategy is to intentionally foul the other team's worst free-throw shooter. Assume that the Dallas Mavericks' center makes $58 \%$ of his
free-throw shots. Calculate the probabilities for the center as shown in parts (a), (b), and $(\mathrm{c}),$ and show that intentionally fouling the Dallas Mavericks' center is a better strategy than intentionally fouling Jerry Stackhouse.

Audrey Fong

Numerade Educator

Problem 37

Visa Card USA studied how frequently young consumers, ages 18 to $24,$ use plastic (debit and credit) cards in making purchases (Associated Press, January 16,2006 ). The results of the study provided the following probabilities.
$\bullet$ The probability that a consumer uses a plastic card when making a purchase is .37
$\bullet$ Given that the consumer uses a plastic card, there is a .19 probability that the consumer is 18 to 24 years old.
$\bullet$ Given that the consumer uses a plastic card, there is a .81 probability that the consumer is more than 24 years old.
U.S. Census Bureau data show that $14 \%$ of the consumer population is 18 to 24 years old.
a. Given the consumer is 18 to 24 years old, what is the probability that the consumer use a plastic card?
b. Given the consumer is over 24 years old, what is the probability that the consumer uses a plastic card?
c. What is the interpretation of the probabilities shown in parts (a) and (b)?
d. Should companies such as Visa, MasterCard, and Discover make plastic cards available to the 18 to 24 year old age group before these consumers have had time to establish a credit history? If no, why? If yes, what restrictions might the companies place on this age group?

Jason Gerber

Numerade Educator

Problem 38

A Morgan Stanley Consumer Research Survey sampled men and women and asked each whether they preferred to drink plain bottled water or a sports drink such as Gatorade or Propel Fitness water (The Atlanta Journal-Constitution, December 28,2005 ). Suppose 200 men and 200 women participated in the study, and 280 reported they preferred plain bottled water. Of the group preferring a sports drink, 80 were men and 40 were women.
Let
$M=$ the event the consumer is a man $W=$ the event the consumer is a woman $B=$ the event the consumer preferred plain bottled water $S=$ the event the consumer preferred sports drink
a. What is the probability a person in the study preferred plain bottled water?
b. What is the probability a person in the study preferred a sports drink?
c. What are the conditional probabilities $P(M | S)$ and $P(W | S) ?$
d. What are the joint probabilities $P(M \cap S)$ and $P(W \cap S) ?$
e. Given a consumer is a man, what is the probability he will prefer a sports drink?
f. Given a consumer is a woman, what is the probability she will prefer a sports drink?
g. Is preference for a sports drink independent of whether the consumer is a man or a woman? Explain using probability information.

Jason Gerber

Numerade Educator

Problem 39

The prior probabilities for events $A_{1}$ and $A_{2}$ are $P\left(A_{1}\right)=.40$ and $P\left(A_{2}\right)=.60 .$ It is also known that $P\left(A_{1} \cap A_{2}\right)=0 .$ Suppose $P\left(B | A_{1}\right)=.20$ and $P\left(B | A_{2}\right)=.05$
a. $\quad$ Are $A_{1}$ and $A_{2}$ mutually exclusive? Explain.
b. $\quad$ Compute $P\left(A_{1} \cap B\right)$ and $P\left(A_{2} \cap B\right)$
c. $\quad$ Compute $P(B)$
d. Apply Bayes' theorem to compute $P\left(A_{1} | B\right)$ and $P\left(A_{2} | B\right)$

Danielle Fairburn

Danielle Fairburn

Numerade Educator

Problem 40

The prior probabilities for events $A_{1}, A_{2},$ and $A_{3}$ are $P\left(A_{1}\right)=.20, P\left(A_{2}\right)=.50,$ and $P\left(A_{3}\right)=$
$.30 .$ The conditional probabilities of event $B$ given $A_{1}, A_{2},$ and $A_{3}$ are $P\left(B | A_{1}\right)=.50$ $P\left(B | A_{2}\right)=.40,$ and $P\left(B | A_{3}\right)=.30$
a. $\quad$ Compute $P\left(B \cap A_{1}\right), P\left(B \cap A_{2}\right),$ and $P\left(B \cap A_{3}\right)$
b. Apply Bayes' theorem, equation (4.19), to compute the posterior probability $P\left(A_{2} | B\right)$.
c. Use the tabular approach to applying Bayes' theorem to compute $P\left(A_{1} | B\right), P\left(A_{2} | B\right)$ and $P\left(A_{3} | B\right)$

Aman Gupta

Numerade Educator

Problem 41

A consulting firm submitted a bid for a large research project. The firm's management initially felt they had a $50-50$ chance of getting the project. However, the agency to which the bid was submitted subsequently requested additional information on the bid. Past experience indicates that for $75 \%$ of the successful bids and $40 \%$ of the unsuccessful bids the agency requested additional information.
a. What is the prior probability of the bid being successful (that is, prior to the request for additional information)?
b. What is the conditional probability of a request for additional information given that the bid will ultimately be successful?
c. Compute the posterior probability that the bid will be successful given a request for additional information.

Bailey Brooks

Numerade Educator

Problem 42

A local bank reviewed its credit card policy with the intention of recalling some of its credit cards. In the past approximately $5 \%$ of cardholders defaulted, leaving the bank unable to collect the outstanding balance. Hence, management established a prior probability of .05 that any particular cardholder will default. The bank also found that the probability of missing a monthly payment is .20 for customers who do not default. Of course, the probability of missing a monthly payment for those who default is 1
a. Given that a customer missed one or more monthly payments, compute the posterior probability that the customer will default.
b. The bank would like to recall its card if the probability that a customer will default is greater than $.20 .$ Should the bank recall its card if the customer misses a monthly payment? Why or why not?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 43

Small cars get better gas mileage, but they are not as safe as bigger cars. Small cars accounted for $18 \%$ of the vehicles on the road, but accidents involving small cars led to 11,898 fatalities during a recent year (Reader's Digest, May 2000). Assume the probability a small car is involved in an accident is $.18 .$ The probability of an accident involving a small car leading to a fatality is .128 and the probability of an accident not involving a small car leading to a fatality is $.05 .$ Suppose you learn of an accident involving a fatality. What is the probability a small car was involved? Assume that the likelihood of getting into an accident is independent of car size.

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 44

The American Council of Education reported that $47 \%$ of college freshmen earn a degree and graduate within five years (Associated Press, May 6,2002 ). Assume that graduation records show women make up $50 \%$ of the students who graduated within five years, but only $45 \%$ of the students who did not graduate within five years. The students who had not graduated within five years either dropped out or were still working on their degrees.
a. $\quad$ Let $A_{1}=$ the student graduated within five years $A_{2}=$ the student did not graduate within five years $W=$ the student is a female student Using the given information, what are the values for $P\left(A_{1}\right), P\left(A_{2}\right), P\left(W | A_{1}\right),$ and
\[
P\left(W | A_{2}\right) ?
\]
b. What is the probability that a female student will graduate within five years?
c. What is the probability that a male student will graduate within five years?
d. Given the preceding results, what are the percentage of women and the percentage of men in the entering freshman class?

Jason Gerber

Numerade Educator

Problem 45

In an article about investment alternatives, Money magazine reported that drug stocks provide a potential for long-term growth, with over $50 \%$ of the adult population of the United States taking prescription drugs on a regular basis. For adults age 65 and older, $82 \%$ take prescription drugs regularly. For adults age 18 to $64,49 \%$ take prescription drugs regularly. The age $18-64$ age group accounts for $83.5 \%$ of the adult population (Statistical Abstract of the United States, 2008 ).
a. What is the probability that a randomly selected adult is 65 or older?
b. Given an adult takes prescription drugs regularly, what is the probability that the adult is 65 or older?

Jason Gerber

Numerade Educator

Problem 46

The Wall Street Journal/Harris Personal Finance poll asked 2082 adults if they owned a home (All Business website, January 23,2008 ). A total of 1249 survey respondents answered Yes. Of the 450 respondents in the $18-34$ age group, 117 responded Yes.
a. What is the probability that a respondent to the poll owned a home?
b. What is the probability that a respondent in the $18-34$ age group owned a home?
c. What is the probability that a respondent to the poll did not own a home?
d. What is the probability that a respondent in the $18-34$ age group did not own a home?

Joshua Hale

Numerade Educator

Problem 47

A financial manager made two new investments-one in the oil industry and one in municipal bonds. After a one-year period, each of the investments will be classified as either successful or unsuccessful. Consider the making of the two investments as an experiment.
a. How many sample points exist for this experiment?
b. Show a tree diagram and list the sample points.
c. $\operatorname{Let} O=$ the event that the oil industry investment is successful and $M=$ the event that the municipal bond investment is successful. List the sample points in $O$ and in $M$
d. List the sample points in the union of the events $(O \cup M)$
e. List the sample points in the intersection of the events $(O \cap M)$
f. Are events $O$ and $M$ mutually exclusive? Explain.

Nick Johnson

Numerade Educator

Problem 48

In early $2003,$ President Bush proposed eliminating the taxation of dividends to shareholders on the grounds that it was double taxation. Corporations pay taxes on the earnings that are later paid out in dividends. In a poll of 671 Americans, TechnoMetrica Market Intelligence found that $47 \%$ favored the proposal, $44 \%$ opposed it, and $9 \%$ were not sure (Investor's Business Daily, January 13,2003 ). In looking at the responses across party lines the poll showed that $29 \%$ of Democrats were in favor, $64 \%$ of Republicans were in favor, and $48 \%$ of Independents were in favor.
a. How many of those polled favored elimination of the tax on dividends?
b. What is the conditional probability in favor of the proposal given the person polled is a Democrat?
c. Is party affiliation independent of whether one is in favor of the proposal?
d. If we assume people's responses were consistent with their own self-interest, which group do you believe will benefit most from passage of the proposal?

Sheryl Ezze

Numerade Educator

Problem 49

A study of 31,000 hospital admissions in New York State found that $4 \%$ of the admissions led to treatment-caused injuries. One-seventh of these treatment-caused injuries resulted in death, and one-fourth were caused by negligence. Malpractice claims were filed in one out of 7.5 cases involving negligence, and payments were made in one out of every two claims.
a. What is the probability a person admitted to the hospital will suffer a treatment-caused injury due to negligence?
b. What is the probability a person admitted to the hospital will die from a treatmentcaused injury?
c. In the case of a negligent treatment-caused injury, what is the probability a malpractice claim will be paid?

Nick Johnson

Numerade Educator

Problem 50

A telephone survey to determine viewer response to a new television show obtained the following data.
a. What is the probability that a randomly selected viewer will rate the new show as average or better?
b. What is the probability that a randomly selected viewer will rate the new show below average or worse?

Audrey Fong

Numerade Educator

Problem 51

The following crosstabulation shows household income by educational level of the head of household (Statistical Abstract of the United States, 2008 ).
a. Develop a joint probability table.
b. What is the probability of a head of household not being a high school graduate?
c. What is the probability of a head of household having a bachelor's degree or more education?
d. What is the probability of a household headed by someone with a bachelor's degree earning $\$ 100,000$ or more?
e. What is the probability of a household having income below $\$ 25,000 ?$
f. What is the probability of a household headed by someone with a bachelor's degree earning less than $\$ 25,000 ?$
g. Is household income independent of educational level?

Bailey Brooks

Numerade Educator

Problem 52

An MBA new-matriculants survey provided the following data for 2018 students.
a. For a randomly selected MBA student, prepare a joint probability table for the experiment consisting of observing the student's age and whether the student applied to one or more schools.
b. What is the probability that a randomly selected applicant is 23 or under?
c. What is the probability that a randomly selected applicant is older than $26 ?$
d. What is the probability that a randomly selected applicant applied to more than one school?

Audrey Fong

Numerade Educator

Problem 53

Refer again to the data from the MBA new-matriculants survey in exercise 52
a. Given that a person applied to more than one school, what is the probability that the person is $24-26$ years old?
b. Given that a person is in the 36 -and-over age group, what is the probability that the person applied to more than one school?
c. What is the probability that a person is $24-26$ years old or applied to more than one school?
d. Suppose a person is known to have applied to only one school. What is the probability that the person is 31 or more years old?
e. Is the number of schools applied to independent of age? Explain.

Audrey Fong

Numerade Educator

Problem 54

An IBD/TIPP poll conducted to learn about attitudes toward investment and retirement (Investor's Business Daily, May 5,2000 ) asked male and female respondents how important they felt level of risk was in choosing a retirement investment. The following joint probability table was constructed from the data provided. "Important" means the respondent said level of risk was either important or very important.
a. What is the probability a survey respondent will say level of risk is important?
b. What is the probability a male respondent will say level of risk is important?
c. What is the probability a female respondent will say level of risk is important?
d. Is the level of risk independent of the gender of the respondent? Why or why not?
e. Do male and female attitudes toward risk differ?

Audrey Fong

Numerade Educator

Problem 55

A large consumer goods company ran a television advertisement for one of its soap products. On the basis of a survey that was conducted, probabilities were assigned to the following events.
$B=$ individual purchased the product $S=$ individual recalls seeing the advertisement $B \cap S=$ individual purchased the product and recalls seeing the advertisement
The probabilities assigned were $P(B)=.20, P(S)=.40,$ and $P(B \cap S)=.12$
a. What is the probability of an individual's purchasing the product given that the individual recalls seeing the advertisement? Does seeing the advertisement increase the probability that the individual will purchase the product? As a decision maker, would you recommend continuing the advertisement (assuming that the cost is reasonable)?
b. Assume that individuals who do not purchase the company's soap product buy from its competitors. What would be your estimate of the company's market share? Would you expect that continuing the advertisement will increase the company's market share? Why or why not?
c. The company also tested another advertisement and assigned it values of $P(S)=.30$ and $P(B \cap S)=.10 .$ What is $P(B | S)$ for this other advertisement? Which advertisement seems to have had the bigger effect on customer purchases?

Audrey Fong

Numerade Educator

Problem 56

Cooper Realty is a small real estate company located in Albany, New York, specializing primarily in residential listings. They recently became interested in determining the likelihood of one of their listings being sold within a certain number of days. An analysis of company sales of 800 homes in previous years produced the following data.
a. If $A$ is defined as the event that a home is listed for more than 90 days before being sold, estimate the probability of $A$
b. If $B$ is defined as the event that the initial asking price is under $\$ 150,000$, estimate the probability of $B$
c. What is the probability of $A \cap B ?$
d. Assuming that a contract was just signed to list a home with an initial asking price of less than $\$ 150,000$, what is the probability that the home will take Cooper Realty more than 90 days to sell?
e. Are events $A$ and $B$ independent?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 57

A company studied the number of lost-time accidents occurring at its Brownsville, Texas, plant. Historical records show that $6 \%$ of the employees suffered lost-time accidents last year. Management believes that a special safety program will reduce such accidents to $5 \%$ during the current year. In addition, it estimates that $15 \%$ of employees who had lost-time accidents last year will experience a lost-time accident during the current year.
a. What percentage of the employees will experience lost-time accidents in both years?
b. What percentage of the employees will suffer at least one lost-time accident over the two-year period?

Jason Gerber

Numerade Educator

Problem 58

A survey showed that $8 \%$ of Internet users age 18 and older report keeping a blog. Referring to the $18-29$ age group as young adults, the survey showed that for bloggers $54 \%$ are young adults and for nonbloggers $24 \%$ are young adults (Pew Internet \& American Life Project, July 19,2006 ).
a. Develop a joint probability table for these data with two rows (bloggers vs. non-bloggers) and two columns (young adults vs. older adults).
b. What is the probability that an Internet user is a young adult?
c. What is the probability that an Internet user keeps a blog and is a young adult?
d. Suppose that in a follow-up phone survey we contact someone who is 24 years old. What is the probability that this person keeps a blog?

Audrey Fong

Numerade Educator

Problem 59

An oil company purchased an option on land in Alaska. Preliminary geologic studies assigned the following prior probabilities.
\[
\begin{aligned}
P(\text { high-quality oil }) &=.50 \\
P(\text { medium-quality oil }) &=.20 \\
P(\text { no oil }) &=.30
\end{aligned}
\]
a. What is the probability of finding oil?
b. After 200 feet of drilling on the first well, a soil test is taken. The probabilities of finding the particular type of soil identified by the test follow.
\[
\begin{aligned}
P(\text { soil } | \text { high-quality oil }) &=.20 \\
P(\text { soil } | \text { medium-quality oil }) &=.80 \\
P(\text { soil } | \text { no oil }) &=.20
\end{aligned}
\]
How should the firm interpret the soil test? What are the revised probabilities, and what is the new probability of finding oil?

Pratyush Raitan

Pratyush Raitan

Numerade Educator

Problem 60

Companies that do business over the Internet can often obtain probability information about website visitors from previous websites visited. The article "Internet Marketing" (Interfaces, March/April 2001) described how clickstream data on websites visited could be used in conjunction with a Bayesian updating scheme to determine the gender of a website visitor. Par Fore created a website to market golf equipment and apparel. Management would like a certain offer to appear for female visitors and a different offer to appear for male visitors. From a sample of past website visits, management learned that $60 \%$ of the visitors to the website ParFore are male and $40 \%$ are female.
a. What is the prior probability that the next visitor to the website will be female?
b. Suppose you know that the current visitor to the website ParFore previously visited the Dillard's website, and that women are three times as likely to visit the Dillard's website as men. What is the revised probability that the current visitor to the website ParFore is female? Should you display the offer that appeals more to female visitors or the one that appeals more to male visitors?

Pratyush Raitan

Pratyush Raitan

Numerade Educator