Chapter4: 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 (www.powerball.com, March 19, 2006). Compute the number of ways the first five numbers can be selected. PART A: 55 balls, 5 are chosen: 55C5 = (55 x 54 x 53 x 52 x 51) / (5 x 4 x 3 x 2 x 1) = 3,478,761 ways PART B: There is 1 way to match the numbers. And we just computed there are 3,478,761 ways for those 5 numbers to be picked. P(match 5) = 1 / (3,478,761) PART C: In addition to match the first 5, you must also match the Powerball. P(match 5 and powerball) = 1/(3,478,761) x 1/(46) = 1 / 146,107,962 P.S. The newer probabilities would be 1 in 11238513 for part B and 1 in 292,201,338 for part C. 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? 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? 26. Data on the 30 largest stock and balanced funds provided one-year and five-year percentage returns for the period ending March 31, 2000 (The Wall Street Journal,April 10, 2000). Suppose we consider a one-year return in excess of 50% to be high and a five-year return in excess of 300% to be high. Nine of the funds had one-year returns in excess of 50%, seven of the funds had five-year returns in excess of 300%, and five of the funds had both one-year returns in excess of 50% and five-year returns in excess of 300%. a. What is the probability of a high one-year return, and what is the probability of a high five-year return? b. What is the probability of both a high one-year return and a high five-year return? c. What is the probability of neither a high one-year return nor a high five-year return? 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.” Reason for Application School School Cost or Quality Convenience Other Totals Enrollment Status Full Time Part Time 421 393 76 400 593 46 890 1039 Totals 821 986 122 1929 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. 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 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. Chapter5 21. The following probability distributions of job satisfaction scores for a sample of information systems (IS) senior executives and IS middle managers range from a low of 1 (very dissatisfied) to a high of 5 (very satisfied). Probability Job Satisfaction IS Senior IS Middle Score Executives Managers 1 .05 .04 2 .09 .10 3 .03 .12 4 .42 .46 5 .41 .28 a. What is the expected value of the job satisfaction score for senior executives? b. What is the expected value of the job satisfaction score for middle managers? c. Compute the variance of job satisfaction scores for executives and middle managers. d. Compute the standard deviation of job satisfaction scores for both probability distributions. e. Compare the overall job satisfaction of senior executives and middle managers 35. A university found that 20% of its students withdraw without completing the introductory statistics course. Assume that 20 students registered for the course. a. Compute the probability that two or fewer will withdraw. b. Compute the probability that exactly four will withdraw. c. Compute the probability that more than three will withdraw. d. Compute the expected number of withdrawals. 43. Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 10 passengers per minute. a. Compute the probability of no arrivals in a one-minute period. b. Compute the probability that three or fewer passengers arrive in a one-minute period. c. Compute the probability of no arrivals in a 15-second period. d. Compute the probability of at least one arrival in a 15-second period 56. A survey conducted by the Bureau of Transportation Statistics (BTS) showed that the average commuter spends about 26 minutes on a one-way door-to-door trip from home to work. In addition, 5% of commuters reported a one-way commute of more than one hour (www.bts.gov, January 12, 2004). a. If 20 commuters are surveyed on a particular day, what is the probability that three will report a one-way commute of more than one hour? b. If 20 commuters are surveyed on a particular day, what is the probability that none will report a one-way commute of more than one hour? c. If a company has 2000 employees, what is the expected number of employees that have a one-way commute of more than one hour? d. If a company has 2000 employees, what is the variance and standard deviation of the number of employees that have a one-way commute of more than one hour? 65. A deck of playing cards contains 52 cards, four of which are aces. What is the probability that the deal of a five-card hand provides: a. A pair of aces? b. Exactly one ace? c. No aces? d. At least one ace? Chapter6 5. The driving distance for the top 100 golfers on the PGA tour is between 284.7 and 310.6 yards (Golfweek, March 29, 2003). Assume that the driving distance for these golfers is uniformly distributed over this interval. a. Give a mathematical expression for the probability density function of driving distance. b. What is the probability the driving distance for one of these golfers is less than 290 yards? c. What is the probability the driving distance for one of these golfers is at least 300 yards? d. What is the probability the driving distance for one of these golfers is between 290 and 305 yards? e. How many of these golfers drive the ball at least 290 yards? 17. For borrowers with good credit scores, the mean debt for revolving and installment accounts is $15,015 (BusinessWeek, March 20, 2006). Assume the standard deviation is $3540 and that debt amounts are normally distributed. a. What isthe probability that the debt for a randomly selected borrower with good credit is more than $18,000? b. What isthe probability that the debt for a randomly selected borrower with good credit is less than $10,000? c. What isthe probability that the debt for a randomly selected borrower with good credit is between $12,000 and $18,000? d. What isthe probability that the debt for a randomly selected borrower with good credit is no more than $14,000? 28. President Bush proposed the elimination of taxes on dividends paid to shareholders on the grounds that they result in double taxation. The earnings used to pay dividends are already taxed to the corporation. A survey on this issue revealed that 47% of Americans favor the proposal. By political party, 64% of Republicans and 29% of Democrats favor the proposal (Investor’s Business Daily, January 13, 2003). Suppose a group of 250 Americans gather to hear a speech about the proposal. a. What is the probability at least half of the group is in favor of the proposal? b. You later find out 150 Republicans and 100 Democrats are present. Now what is your estimate of the expected number in favor of the proposal? c. Now that you know the composition of the group, do you expect a speaker in favor of the proposal will be better received than one against the proposal? 35. The time between arrivals of vehicles at a particular intersection follows an exponential probability distribution with a mean of 12 seconds. a. Sketch this exponential probability distribution. b. What is the probability that the arrival time between vehicles is 12 seconds or less? c. What is the probability that the arrival time between vehicles is 6 seconds or less? d. What is the probability of 30 or more seconds between vehicle arrivals? 4O. The U.S. Bureau of Labor Statistics reports that the average annual expenditure on food and drink for all families is $5700 (Money, December 2003). Assume that annual expenditure on food and drink is normally distributed and that the standard deviation is $1500. a. What is the range of expenditures of the 10% of families with the lowest annual spending on food and drink? b. What percentage of families spend more than $7000 annually on food and drink? c. What is the range of expenditures for the 5% of families with the highest annual spending on food and drink? 50. A blackjack player at a Las Vegas casino learned that the house will provide a free room if play is for four hours at an average bet of $50. The player’s strategy provides a probability of .49 of winning on any one hand, and the player knows that there are 60 hands per hour. Suppose the player plays for four hours at a bet of $50 per hand. a. What is the player’s expected payoff? b. What is the probability the player loses $1000 or more? c. What is the probability the player wins? d. Suppose the player starts with $1500. What is the probability of going broke?