## probability

A package of fifteen Ping-Pong balls contains two defectives. An inspection procedure to check on the quality of the package is to select randomly three Ping-Pong balls for inspection and to reject the entire package if any of the three is defective. What is the probability that this package will be rejected?

6. Assume that in a marketing study it has been found that on a fall Saturday afternoon 20% of adult men watch sports programming on network A from 1:00 P.M. to 2:00 P.M., 25% watch on network C from 2:00 P.M. to 3:00 P.M., and 10% watch on both networks. For simplicity assume that viewers of these programs are sure to see advertisements if they are watching a program. If Chips ‘n’ Fish Restaurants advertise only on network C during the sports programming segment, (a) what is the probability that the ad will be seen by a randomly chosen adult man? (b) What would be the probability if they advertise on both networks? (c) If they advertise on both networks, what is the probability that an ad will be seen only on network A?

7. You fly from Philadelphia to San Francisco with a connection in Dallas. The probability that your flight from Philadelphia to Dallas arrives on time is 0.85. If you arrive on time, then the probability that your luggage makes the connection to San Francisco is 0.95. If you are delayed, then the probability that your luggage makes the connection with you is 0.6. In either case, you make the flight.

(a) What is the probability that your luggage is there to meet you in San Francisco?

(b) If your luggage is not there to meet you, what is the probability that you were late arriving in Dallas?

8. A law firm takes cases on a contingent fee basis. If the case goes to trial, the firm expects to earn $25,000 as part of the settlement if it wins and nothing if it does not. The firm wins one-third of the cases that go to trial. If the case does not go to trial, the firm earns nothing. Half of the cases do not go to trial. What is the probability distribution of this firm’s earnings?

9. Beth, Seth’s sister, sells ice cream in the same area as he, and her daily sales follow the same pattern as his, namely, a mean of 300 per day and a standard deviation of 120 per day. However, for her pay, she receives $10 per day from her employer plus $.08 commission on each ice cream sale. Find (a) her expected daily profit, and (b) the standard deviation of her daily profit.

10. Historically a bank expects about 4% of its borrowers to default (not repay). The bank currently has 200 loans outstanding. The bank also has enough reserves to cover losses if 20 of these loans were to default. Will these reserves be enough? Explain why or why not.

11. An insurance company found that 2.5% of male drivers between the ages of 18 and 25 are involved in serious accidents annually. Assume that every such accident costs the company $65,000 and that a driver can only have one of these accidents in a year.

(a) If the company charges $2,500 for such coverage, what is the probability that it loses money on a single policy?

(b) Suppose that the company writes 1,000 such policies to a collection of drivers. What is the probability that the company makes a profit on these policies? Assume that the drivers don’t run into each other and behave independently.

(c) Does the difference between the probabilities of parts (a) and (b) explain how insurance companies stay in business? Large auto insurers are certainly profitable. One report, for example, claims that Allstate pays out less than $0.50 in accident claims for every dollar collected in premiums. (Business Week, 5/1/2006)

12. In operations management, you learn that intermittent systems produce a variety of products one at a time (or in batches) to customer order. These systems contrast with continuous systems, in which a large number of homogeneous products are produced. One feature of intermittent systems is that new orders arrive at irregular intervals. For example, orders for business letterhead may come into the Ben Franklin Print Shop at the rate of .5 per day. Find the probability of (a) at least 2 orders for letterhead arriving tomorrow, (b) no orders in the next 2 days. (c) Find the expected number of orders in the next 30 working days. (d) What is the probability that the time between two orders is less than one day?

13. Industrial accidents can be very costly to both workers and employers. Prior to a safety campaign at the American Banana Company there was an average of 8 accidents per month. In the month after the campaign there was one accident. If the safety campaign had no effect on the average, what would be the probability of at most one accident? Do you believe the campaign had a positive effect? Why?

14. A cafeteria vending machine dispenses 6 ounces of coffee per cup, on the average, in such a way that the amount dispensed per cup is a normally distributed random variable. How fine should the machine be tuned; that is, to what level should the standard deviation be set, so that 99% of the cups are filled with at least 5.9 ounces of coffee?

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15. In screening applicants for a position, testing is one device used, in addition to interviews and background investigations. Such tests often focus on mechanical aptitudes and skills, although for some positions, tests may focus on personality psychology. One such test for manual dexterity administered to a large number of applicants showed that the time required for completion was approximately normally distributed with a mean of 12 minutes and with a standard deviation of 1.5 minutes. Based on this information, (a) what proportion of applicants finish in less than 14 minutes? (b) If only the best 25% of the applicants (using the dexterity test as the criterion of best) are to be passed through this screening stage, within what time must an applicant complete this test in order to pass?