It is not blue. d. It is both red and brown. b.

Number Surveyed Number Responding “Poor” Site 1 192 48 Site 2 248 80 Let A be the event the worker comes from Site 1 and B be the event the response is “poor.” Com- pute PA, PB, and . 4.25 Refer to Exercise 4.23 a. Are events A and B independent?

b. Find

and . Are they equal? H.R. 4.26 A large corporation has spent considerable time developing employee performance rating scales to evaluate an employee’s job performance on a regular basis, so major adjustments can be made when needed and employees who should be considered for a “fast track” can be isolated. Keys to this latter determination are ratings on the ability of an employee to perform to his or her capabilities and on his or her formal training for the job. Formal Training Workload Capacity None Little Some Extensive Low .01 .02 .02 .04 Medium .05 .06 .07 .10 High .10 .15 .16 .22 The probabilities for being placed on a fast track are as indicated for the 12 categories of work- load capacity and formal training. The following three events A, B, and C are defined: A: An employee works at the high-capacity level B: An employee falls into the highest extensive formal training category C: An employee has little or no formal training and works below high capacity

a. Find PA, PB, and PC. b. Find , , and .

c. Find , ,

and . Bus. 4.27 The utility company in a large metropolitan area finds that 70 of its customers pay a given monthly bill in full.

a. Suppose two customers are chosen at random from the list of all customers. What is

the probability that both customers will pay their monthly bill in full? b. What is the probability that at least one of them will pay in full? 4.28 Refer to Exercise 4.27. A more detailed examination of the company records indicates that 95 of the customers who pay one monthly bill in full will also pay the next monthly bill in full; only 10 of those who pay less than the full amount one month will pay in full the next month.

a. Find the probability that a customer selected at random will pay two consecutive

months in full.

b. Find the probability that a customer selected at random will pay neither of two

consecutive months in full.

c. Find the probability that a customer chosen at random will pay exactly one month in full.

4.5 Bayes’ Formula Bus. 4.29 Of a finance company’s loans, 1 are defaulted not completely repaid. The company routinely runs credit checks on all loan applicants. It finds that 30 of defaulted loans went to poor risks, 40 to fair risks, and 30 to good risks. Of the nondefaulted loans, 10 went to poor risks, 40 to fair risks, and 50 to good risks. Use Bayes’ Formula to calculate the probability that a poor-risk loan will be defaulted. 4.30 Refer to Exercise 4.29. Show that the posterior probability of default, given a fair risk, equals the prior probability of default. Explain why this is a reasonable result. PB 傽 C PA 傽 C PA 艛 B PB | C PB | B PA|B PB | A PB | A PA 傽 B