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4.2 How many ways can three items be selected from a group of six items? Use the letters A, B, C, D, E, and F to identify the items, and list each of the different combinations of three items.
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4.3 How many permutations of three items can be selected from a group of six? Use the letters A, B, C, D, E, and F to identify the items, and list each of the permutations of items B, D, and F.
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4.4 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?
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4.5 Suppose an experiment has five equally likely outcomes: E1, E2, E3, E4, E5. Assign probabilities to each outcome and show that the requirements in equations (4.3) and (4.4) are satisfied. What method did you use?
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4.6 An experiment with three outcomes has been repeated 50 times, and it was learned that E1 occurred 20 times, E2 occurred 13 times, and E3 occurred 17 times. Assign probabilities to the outcomes. What method did you use?
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4.7 A decision maker subjectively assigned the following probabilities to the four outcomes of an experiment: P(E1) = .10, P(E2) = .15, P(E3) = .40, and P(E4) = .20. Are these probability assignments valid? Explain.
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4.8 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.
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4.9 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?
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4.10
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4.11 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.
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4.12 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 (http://www.powerball.com, 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?
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4.13 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....
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4.14 An experiment has four equally likely outcomes: E1, E2, E3, and E4.
a. What is the probability that E2 occurs?
b. What is the probability that any two of the outcomes occur (e.g., E1 or E3)?
c. What is the probability that any three of the outcomes occur (e.g., E1 or E2 or E4)?
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4.15 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 (c).
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4.16 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, and 12) and only five possible odd values (3, 5, 7, 9, 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?
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4.17 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?
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4.18
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4.19 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?
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4.20 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).
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4.21
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4.22 Suppose that we have a sample space with five equally likely experimental outcomes: E1, E2, E3, E4, E5. Let...
a. Find P(A), P(B), and P(C).
b. Find P(A ∪ B). Are A and B mutually exclusive?
c. Find Ac, Cc, P(Ac), and P(Cc).
d. Find A ∪ Bc and P(A ∪ Bc).
e. Find P(B ∪ C).
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4.23 Suppose that we have a sample space S = {E1, E2, E3, E4, E5, E6, E7}, where E1, E2, … , E7 denote the sample points. The following probability assignments apply: P(E1) = .05, P(E2) = .20, P(E3) = .20, P(E4) = .25, P(E5) = .15, P(E6) = .10, and P(E7) = .05. Let...
a. Find P(A), P(B), and P(C).
b. Find A ∪ B and P(A ∪ B).
c. Find A ∩ B and P(A ∩ B).
d. Are events A and C mutually exclusive?
e. Find Bc and P(Bc).
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4.24 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?
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4.25 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...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 2019 home?
b. What is the probability both young adults selected are living on their own (neither is living in their parents’ home)?
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4.26
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4.27
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4.28 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?
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4.29 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 ∩ 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?
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4.30 Suppose that we have two events, A and B, with P(A) = .50, P(B) = .60, and P(A ∩ B) = .40.
a. Find P(A | B).
b. Find P(B | A).
c. Are A and B independent? Why or why not?
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4.31 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 ∩ B)?
b. What is P(A | B)?
c. 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?
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4.32
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4.33 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.
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4.34
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4.35
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4.36
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4.37 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.
• The probability that a consumer uses a plastic card when making a purchase is .37.• Given that the consumer uses a plastic card, there is a .19 probability that the consumer is 18 to 24 years old.• 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?
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4.38 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...
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 ∩ S) and P(W ∩ 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.
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4.39 The prior probabilities for events A1 and A2 are P(A1) = .40 and P(A2) = .60. It is also known that P(A1 ∩ A2) = 0. Suppose P(B | A1) = .20 and P(B | A2) = .05.
a. Are A1 and A2 mutually exclusive? Explain.
b. Compute P(A1 ∩ B) and P(A2 ∩ B).
c. Compute P(B).
d. Apply Bayes’ theorem to compute P(A1 | B) and P(A2 | B).
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4.40 The prior probabilities for events A1, A2, and A3 are P(A1) = .20, P(A2) = .50, and P(A3) = .30. The conditional probabilities of event B given A1, A2, and A3 are P(B | A1) = .50, P(B | A2) = .40, and P(B | A3) = .30.
a. Compute P(B ∩ A1), P(B ∩ A2), and P(B ∩ A3).
b. Apply Bayes’theorem, equation (4.19), to compute the posterior probability P(A2 | B).
c. Use the tabular approach to applying Bayes’ theorem to compute P(A1 | B), P(A2 | B), and P(A3 | B).
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4.41 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.
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4.42 Alocal 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?
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4.43 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.
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4.44 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. Let A1 = the student graduated within five yearsA2 = the student did not graduate within five yearsW = the student is a female studentUsing the given information, what are the values for P(A1), P(A2), P(W | A1), and P(W | A2)?
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?
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4.45
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4.46
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4.47 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. 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 ∪ M).
e. List the sample points in the intersection of the events (O ∩ M).
f. Are events O and M mutually exclusive? Explain.
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4.48 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’sBusiness 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?
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4.49 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 treatment-caused injury?
c. In the case of a negligent treatment-caused injury, what is the probability a malpractice claim will be paid?
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4.50 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?
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4.51
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4.52 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?
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4.53 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.
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4.54 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?
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4.55 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 productS = individual recalls seeing the advertisementB ∩ S = individual purchased the product and recalls seeing the advertisementThe probabilities assigned were P(B) = .20, P(S) = .40, and P(B ∩ 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 ∩ S) = .10. What is P(B | S) for this other advertisement? Which advertisement seems to have had the bigger effect on customer purchases?
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4.56 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 ∩ 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?
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4.57 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?
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4.58
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4.59 An oil company purchased an option on land in Alaska. Preliminary geologic studies assigned the following prior probabilities....
a. What is the probability of finding oil?...How should the firm interpret the soil test? What are the revised probabilities, and what is the new 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.
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4.60 Companies that do business over the Internet can often obtain probability information about Web site visitors from previous Web sites visited. The article “Internet Marketing” (Interfaces, March/April 2001) described how clickstream data on Web sites visited could be used in conjunction with a Bayesian updating scheme to determine the gender of a Web site visitor. Par Fore created a Web site 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 Web site visits, management learned that 60% of the visitors to ParFore.com are male and 40% are female.
a. What is the prior probability that the next visitor to the Web site will be female?
b. Suppose you know that the current visitor to ParFore.com previously visited the Dillard’s Web site, and that women are three times as likely to visit the Dillard’s Web site as men. What is the revised probability that the current visitor to ParFore.com is female? Should you display the offer that appeals more to female visitors or the one that appeals more to male visitors?
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