# 1. An analyst estimates that the probability of default on a seven-year

AA-rated bond is 0.60 while that on a seven-year A-rated bond is 0.40. The probability that they will both defaults is 0.44.

A. What is the probability that at least one of the bonds defaults? (round your answers to 2 decimal places)

B. What is the probability that neither the seven -year AA-rated bond nor the seven-year A-rated bond defaults? (round your answers to 2 decimal places)

C. Given that the seven -year AA-rated bond defaults, what is the probability that seven- year A rated bond also defaults? (round your answers to 2 decimal places)

2. A marketing firm is considering making up to three new hires. Given its specific needs, management feels that there is a 55% chance of hiring at least two candidates. There is only a 9% chance that it will not make any hires and a 15% chance that it will make all three hires.

A. What is the probability that the firm will make at least one hire? (round your answers to 2 decimal places)

B. Find the expected value and the standard deviation of the number of hires. (round intermediate calculations to at least 4 decimals places. round your final answers to 2 decimal places)

3. Organizers of an outdoor summer concert in Toronto are concerned about the weather conditions on the day of the concert. They will make a profit of $23,000 on a clear day and $15,000 on a cloudy day. They will take a loss of $5000 if it rains. The weather channel has predicted a 57% chance of the rain on the day of the concert. Calculate the expected profit from the concert if the likelihood is 11% that will be sunny and 32% that it will be cloudy.

4. An investor is considers investing $12000 in the stock market. He believes that the probability is 0.23 economy will improve, 0.41 that it will stay the same, and 0.36 that it will deteriorate. Further, if the economy improves, he expects his investment to grow to $21,000, but it can also go down to 9000 if the economy deteriorates. If the economy stays the same, his investment will stay at $12,000. What is expected value of his investment?

5. The state police are trying to crack down on speeding on particular portion of the Massachusetts Turnpike. To aid in this pursuit, they have purchased a new radar gun that promises greater consistency and reliability. Specifically, the gun advertises + one-mile-per-hour accuracy 95% of the time; that is there is a 0.95 probability that gun will detect a speeder if the driver is actually speeding. Assume there is a 3% chance that the gun erroneously detects a speeder even when the driver is below the speed limit. Suppose that 92% of the drivers drive below the speed limit on this stretch of the Massachusetts Turnpike.

a. What is the probability that the gun detects speeding and the driver was speeding? (round your answers to 4 decimal places.)

b. What is the probability that the gun detects speeding and the driver was not speeding? (round your answers to 4 decimal places.)

c. Suppose the police stop a driver because the gun detects speeding. What is the probability that the driver was actually driving below the speed limit? (round your answers to 4 decimal places.)

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