2-14 A student taking Management Science 301 at East Haven University will receive one of the five possible grades for the course: A, B, C, D, or F. The distribution of grades over the past 2 years is as follows:

If this past distribution is a good indicator of future grades, what is the probability of a student receiving a C in the course?

2-15 A silver dollar is flipped twice. Calculate the probability of each of the following occurring:

1. a head on the first flip

2. a tail on the second flip given that the first toss was a head

3. two tails

4. a tail on the first and a head on the second

5. a tail on the first and a head on the second or a head on the first and a tail on the second

6. at least one head on the two flips

2-16 An urn contains 8 red chips, 10 green chips, and 2 white chips. A chip is drawn and replaced, and then a second chip drawn. What is the probability of

1. a white chip on the first draw?

2. a white chip on the first draw and a red on the second?

3. two green chips being drawn?

4. a red chip on the second given that a white chip was drawn on the first?

2-17 Evertight, a leading manufacturer of quality nails, produces 1-, 2-, 3-, 4-, and 5-inch nails for various uses. In the production process, if there is an overrun or the nails are slightly defective, they are placed in a common bin. Yesterday, 651 of the 1-inch nails, 243 of the 2-inch nails, 41 of the 3-inch nails, 451 of the 4-inch nails, and 333 of the 5-inch nails were placed in the bin.

1. What is the probability of reaching into the bin and getting a 4-inch nail?

2. What is the probability of getting a 5-inch nail?

3. If a particular application requires a nail that is 3 inches or shorter, what is the probability of getting a nail that will satisfy the requirements of the application?

2-18 Last year, at Northern Manufacturing Company, 200 people had colds during the year. One hundred fifty-five people who did no exercising had colds, and the remainder of the people with colds were involved in a weekly exercise program. Half of the 1,000 employees were involved in some type of exercise.

1. What is the probability that an employee will have a cold next year?

2. Given that an employee is involved in an exercise program, what is the probability that he or she will get a cold next year?

3. What is the probability that an employee who is not involved in an exercise program will get a cold next year?

4. Are exercising and getting a cold independent events? Explain your answer.

2-19 The Springfield Kings, a professional basketball team, has won 12 of its last 20 games and is expected to continue winning at the same percentage rate. The team’s ticket manager is anxious to attract a large crowd to tomorrow’s game but believes that depends on how well the Kings perform tonight against the Galveston Comets. He assesses the probability of drawing a large crowd to be 0.90 should the team win tonight. What is the probability that the team wins tonight and that there will be a large crowd at tomorrow’s game?

2-20 David Mashley teaches two undergraduate statistics courses at Kansas College. The class for Statistics 201 consists of 7 sophomores and 3 juniors. The more advanced course, Statistics 301, has 2 sophomores and 8 juniors enrolled. As an example of a business sampling technique, Professor Mashley randomly selects, from the stack of Statistics 201 registration cards, the class card of one student and then places that card back in the stack. If that student was a sophomore, Mashley draws another card from the Statistics 201 stack; if not, he randomly draws a card from the Statistics 301 group. Are these two draws independent events? What is the probability of

1. a junior’s name on the first draw?

2. a junior’s name on the second draw given that a sophomore’s name was drawn first?

3. a junior’s name on the second draw given that a junior’s name was drawn first?

4. a sophomore’s name on both draws?

5. a junior’s name on both draws?

6. one sophomore’s name and one junior’s name on the two draws, regardless of order drawn?

2-21 The oasis outpost of Abu Ilan, in the heart of the Negev desert, has a population of 20 Bedouin tribesmen and 20 Farima tribesmen. El Kamin, a nearby oasis, has a population of 32 Bedouins and 8 Farima. A lost Israeli soldier, accidentally separated from his army unit, is wandering through the desert and arrives at the edge of one of the oases. The soldier has no idea which oasis he has found, but the first person he spots at a distance is a Bedouin. What is the probability that he has wandered into Abu Ilan? What is the probability that he is in El Kamin?

2-22 The lost Israeli soldier mentioned in Problem 2-21 decides to rest for a few minutes before entering the desert oasis he has just found. Closing his eyes, he dozes off for 15 minutes, wakes, and walks toward the center of the oasis. The first person he spots this time he again recognizes as a Bedouin. What is the posterior probability that he is in El Kamin?

2-23 Ace Machine Works estimates that the probability its lathe tool is properly adjusted is 0.8. When the lathe is properly adjusted, there is a 0.9 probability that the parts produced pass inspection. If the lathe is out of adjustment, however, the probability of a good part being produced is only 0.2. A part randomly chosen is inspected and found to be acceptable. At this point, what is the posterior probability that the lathe tool is properly adjusted?

2-24 The Boston South Fifth Street Softball League consists of three teams: Mama’s Boys, team 1; the Killers, team 2; and the Machos, team 3. Each team plays the other teams just once during the season. The win–loss record for the past 5 years is as follows:

Each row represents the number of wins over the past 5 years. Mama’s Boys beat the Killers 3 times, beat the Machos 4 times, and so on.

1. What is the probability that the Killers will win every game next year?

2. What is the probability that the Machos will win at least one game next year?

3. What is the probability that Mama’s Boys will win exactly one game next year?

4. What is the probability that the Killers will win fewer than two games next year?

2-25 The schedule for the Killers next year is as follows (refer to Problem 2-24):

• Game 1: The Machos

• Game 2: Mama’s Boys

1. What is the probability that the Killers will win their first game?

2. What is the probability that the Killers will win their last game?

3. What is the probability that the Killers will break even—win exactly one game?

4. What is the probability that the Killers will win every game?

5. What is the probability that the Killers will lose every game?

6. Would you want to be the coach of the Killers?

2-26 The Northside rifle team has two markspersons, Dick and Sally. Dick hits a bull’s-eye 90% of the time, and Sally hits a bull’s-eye 95% of the time.

1. What is the probability that either Dick or Sally or both will hit the bull’s-eye if each takes one shot?

2. What is the probability that Dick and Sally will both hit the bull’s-eye?

3. Did you make any assumptions in answering the preceding questions? If you answered yes, do you think that you are justified in making the assumption(s)?

2-27 In a sample of 1,000 representing a survey from the entire population, 650 people were from Laketown, and the rest of the people were from River City. Out of the sample, 19 people had some form of cancer. Thirteen of these people were from Laketown.

1. Are the events of living in Laketown and having some sort of cancer independent?

2. Which city would you prefer to live in, assuming that your main objective was to avoid having cancer?

2-28 Compute the probability of “loaded die given that a 3 was rolled,” as shown in the example in Section 2.3, this time using the general form of Bayes’ Theorem from Equation 2-5.

2-29 Which of the following are probability distributions? Why?

1. RANDOM VARIABLE X	PROBABILITY
   2	0.1
   $\mathrm{-}1$	0.2
   0	0.3
   1	0.25
   2	0.15

2. RANDOM VARIABLE Y	PROBABILITY
   1	1.1
   1.5	0.2
   2	0.3
   2.5	0.25
   3	$\mathrm{-}1.25$

3. RANDOM VARIABLE Z	PROBABILITY
   1	0.1
   2	0.2
   3	0.3
   4	0.4
   5	0.0

2-30 Harrington Health Food stocks 5 loaves of Neutro-Bread. The probability distribution for the sales of Neutro-Bread is listed in the following table. How many loaves will Harrington sell on average?

2-31 What are the expected value and variance of the following probability distribution?

2-32 There are 10 questions on a true–false test. A student feels unprepared for this test and randomly guesses the answer for each of these.

1. What is the probability that the student gets exactly 7 correct?

2. What is the probability that the student gets exactly 8 correct?

3. What is the probability that the student gets exactly 9 correct?

4. What is the probability that the student gets exactly 10 correct?

5. What is the probability that the student gets more than 6 correct?

2-33 Gary Schwartz is the top salesman for his company. Records indicate that he makes a sale on 70% of his sales calls. If he calls on four potential clients, what is the probability that he makes exactly 3 sales? What is the probability that he makes exactly 4 sales?

2-34 If 10% of all disk drives produced on an assembly line are defective, what is the probability that there will be exactly one defect in a random sample of 5 of these? What is the probability that there will be no defects in a random sample of 5?

2-35 Trowbridge Manufacturing produces cases for personal computers and other electronic equipment. The quality control inspector for this company believes that a particular process is out of control. Normally, only 5% of all cases are deemed defective due to discolorations. If 6 such cases are sampled, what is the probability that there will be 0 defective cases if the process is operating correctly? What is the probability that there will be exactly 1 defective case?

2-36 Refer to the Trowbridge Manufacturing example in Problem 2-35. The quality control inspection procedure is to select 6 items, and if there are 0 or 1 defective cases in the group of 6, the process is said to be in control. If the number of defects is more than 1, the process is out of control. Suppose that the true proportion of defective items is 0.15. What is the probability that there will be 0 or 1 defects in a sample of 6 if the true proportion of defects is 0.15?

2-37 An industrial oven used to cure sand cores for a factory manufacturing engine blocks for small cars is able to maintain fairly constant temperatures. The temperature range of the oven follows a normal distribution with a mean of $450°\text{F}$ and a standard deviation of $25°\text{F}.$ Leslie Larsen, president of the factory, is concerned about the large number of defective cores that have been produced in the past several months. If the oven gets hotter than $475°\text{F},$ the core is defective. What is the probability that the oven will cause a core to be defective? What is the probability that the temperature of the oven will range from $460°$ to $470°\text{F}?$

2-38 Steve Goodman, production foreman for the Florida Gold Fruit Company, estimates that the average sale of oranges is 4,700 and the standard deviation is 500 oranges. Sales follow a normal distribution.

1. What is the probability that sales will be greater than 5,500 oranges?

2. What is the probability that sales will be greater than 4,500 oranges?

3. What is the probability that sales will be less than 4,900 oranges?

4. What is the probability that sales will be less than 4,300 oranges?

2-39 Susan Williams has been the production manager of Medical Suppliers, Inc., for the past 17 years. Medical Suppliers, Inc., is a producer of bandages and arm slings. During the past 5 years, the demand for No-Stick bandages has been fairly constant. On average, sales have been about 87,000 packages of No-Stick. Susan has reason to believe that the distribution of No-Stick follows a normal curve, with a standard deviation of 4,000 packages. What is the probability that sales will be less than 81,000 packages?

2-40 Armstrong Faber produces a standard number-two pencil called Ultra-Lite. Since Chuck Armstrong started Armstrong Faber, sales have grown steadily. With the increase in the price of wood products, however, Chuck has been forced to increase the price of the Ultra-Lite pencils. As a result, the demand for Ultra-Lite has been fairly stable over the past 6 years. On average, Armstrong Faber has sold 457,000 pencils each year. Furthermore, 90% of the time sales have been between 454,000 and 460,000 pencils. It is expected that the sales follow a normal distribution with a mean of 457,000 pencils. Estimate the standard deviation of this distribution. (Hint: Work backward from the normal table to find Z. Then apply Equation 2-13.)

2-41 The time to complete a construction project is normally distributed with a mean of 60 weeks and a standard deviation of 4 weeks.

1. What is the probability the project will be finished in 62 weeks or less?

2. What is the probability the project will be finished in 66 weeks or less?

3. What is the probability the project will take longer than 65 weeks?

2-42 A new integrated computer system is to be installed worldwide for a major corporation. Bids on this project are being solicited, and the contract will be awarded to one of the bidders. As a part of the proposal for this project, bidders must specify how long the project will take. There will be a significant penalty for finishing late. One potential contractor determines that the average time to complete a project of this type is 40 weeks with a standard deviation of 5 weeks. The time required to complete this project is assumed to be normally distributed.

1. If the due date of this project is set at 40 weeks, what is the probability that the contractor will have to pay a penalty (i.e., the project will not be finished on schedule)?

2. If the due date of this project is set at 43 weeks, what is the probability that the contractor will have to pay a penalty (i.e., the project will not be finished on schedule)?

3. If the bidder wishes to set the due date in the proposal so that there is only a 5% chance of being late (and consequently only a 5% chance of having to pay a penalty), what due date should be set?

2-43 Patients arrive at the emergency room of Costa Valley Hospital at an average of 5 per day. The demand for emergency room treatment at Costa Valley follows a Poisson distribution.

1. Using Appendix C, compute the probability of exactly 0, 1, 2, 3, 4, and 5 arrivals per day.

2. What is the sum of these probabilities, and why is the number less than 1?

2-44 Using the data in Problem 2-43, determine the probability of more than 3 visits for emergency room service on any given day.

2-45 Cars arrive at Carla’s Muffler Shop for repair work at an average of 3 per hour, following an exponential distribution.

1. What is the expected time between arrivals?

2. What is the variance of the time between arrivals?

2-46 A particular test for the presence of steroids is to be used after a professional track meet. If steroids are present, the test will accurately indicate this 95% of the time. However, if steroids are not present, the test will indicate this 90% of the time (so it is wrong 10% of the time and predicts the presence of steroids). Based on past data, it is believed that 2% of the athletes do use steroids. This test is administered to one athlete, and the test is positive for steroids. What is the probability that this person actually used steroids?

2-47 Market Researchers, Inc., has been hired to perform a study to determine if the market for a new product will be good or poor. In similar studies performed in the past, whenever the market actually was good, the market research study indicated that it would be good 85% of the time. On the other hand, whenever the market actually was poor, the market study incorrectly predicted it would be good 20% of the time. Before the study is performed, it is believed there is a 70% chance the market will be good. When Market Researchers, Inc., performs the study for this product, the results predict the market will be good. Given the results of this study, what is the probability that the market actually will be good?

2-48 Policy Pollsters is a market research firm specializing in political polls. Records indicate in past elections, when a candidate was elected, Policy Pollsters had accurately predicted this 80% of the time and was wrong 20% of the time. Records also show, for losing candidates, Policy Pollsters accurately predicted they would lose 90% of the time and was wrong only 10% of the time. Before the poll is taken, there is a 50% chance of winning the election. If Policy Pollsters predicts a candidate will win the election, what is the probability that the candidate will actually win? If Policy Pollsters predicts that a candidate will lose the election, what is the probability that the candidate will actually lose?

2-49 Burger City is a large chain of fast-food restaurants specializing in gourmet hamburgers. A mathematical model is now used to predict the success of new restaurants based on location and demographic information for that area. In the past, 70% of all restaurants that were opened were successful. The mathematical model has been tested in the existing restaurants to determine how effective it is. For the restaurants that were successful, 90% of the time the model predicted they would be, while 10% of the time the model predicted a failure. For the restaurants that were not successful, when the mathematical model was applied, 20% of the time it incorrectly predicted a successful restaurant, while 80% of the time it was accurate and predicted an unsuccessful restaurant. If the model is used on a new location and predicts the restaurant will be successful, what is the probability that it actually is successful?

2-50 A mortgage lender attempted to increase its business by marketing its subprime mortgage. This mortgage is designed for people with a less-than-perfect credit rating, and the interest rate is higher to offset the extra risk. In the past year, 20% of these mortgages resulted in foreclosure as customers defaulted on their loans. A new screening system has been developed to determine whether to approve customers for the subprime loans. When the system is applied to a credit application, the system will classify the application as “Approve for loan” or “Reject for loan.” When this new system was applied to recent customers who had defaulted on their loans, 90% of these customers were classified as “Reject.” When this same system was applied to recent loan customers who had not defaulted on their loan payments, 70% of these customers were classified as “Approve for loan.”

1. If a customer did not default on a loan, what is the probability that the rating system would have classified the applicant in the reject category?

2. If the rating system had classified the applicant in the reject category, what is the probability that the customer would not default on a loan?

2-51 Use the F table in Appendix D to find the value of F for the upper 5% of the F distribution with

1. ${\text{df}}_1=8,{\text{ df}}_2=7$

2. ${\text{df}}_1=5,{\text{ df}}_2=10$

3. ${\text{df}}_1=3,{\text{ df}}_2=5$

4. ${\text{df}}_1=10,{\text{ df}}_2=4$

2-52 Use the F table in Appendix D to find the value of F for the upper 1% of the F distribution with

1. ${\text{df}}_1=15,{\text{ df}}_2=6$

2. ${\text{df}}_1=12,{\text{ df}}_2=8$

3. ${\text{df}}_1=3,{\text{ df}}_2=5$

4. ${\text{df}}_1=9,{\text{ df}}_2=7$

2-53 For each of the following F values, determine whether the probability indicated is greater than or less than 5%:

1. $P\left(F_{3,4}>6.8\right)$

2. $P\left(F_{7,3}>3.6\right)$

3. $P\left(F_{20,20}>2.6\right)$

4. $P\left(F_{7,5}>5.1\right)$

5. $P\left(F_{7,5}<5.1\right)$

2-54 For each of the following F values, determine whether the probability indicated is greater than or less than 1%:

1. $P\left(F_{5,4}>14\right)$

2. $P\left(F_{6,3}>30\right)$

3. $P\left(F_{10,12}>4.2\right)$

4. $P\left(F_{2,3}>35\right)$

5. $P\left(F_{2,3}<35\right)$

2-55 Nite Time Inn has a toll-free telephone number so that customers can call at any time to make a reservation. A typical call takes about 4 minutes to complete, and the time required follows an exponential distribution. Find the probability that a call takes

1. 3 minutes or less.

2. 4 minutes or less.

3. 5 minutes or less.

4. longer than 5 minutes.

2-56 During normal business hours on the east coast, calls to the toll-free reservation number of the Nite Time Inn arrive at a rate of 5 per minute. It has been determined that the number of calls per minute can be described by the Poisson distribution. Find the probability that in the next minute, the number of calls arriving will be

1. exactly 5.

2. exactly 4.

3. exactly 3.

4. exactly 6.

5. less than 2.

2-57 In the Arnold’s Muffler example for the exponential distribution in this chapter, the average rate of service was given as 3 per hour, and the times were expressed in hours. Convert the average service rate to the number per minute and convert the times to minutes. Find the probabilities that the service times will be less than ${}^1{/}_2$ hour, ${}^1{/}_3$ hour, and ${}^2{/}_3$ hour. Compare these probabilities to the probabilities found in the example.