|Question||Steve and Leroy buy antique paintings at an art gallery in Fresno, California. Eighty percent of the paintings that are sold at the gallery are fakes, and the rest are genuine. After a painting is purchased, it will be carefully analyzed, and then everybody will know for certain whether it is genuine or a fake. A genuine antique is worth $1,000. A fake is worthless. Before they place their bids, buyers are allowed to inspect the paintings briefly and then must place their bids. Because they are allowed only a brief inspection, Steve and Leroy each try to guess whether the paintings are fakes by smelling them. Steve finds that if a painting fails his sniff test, then it is certainly a fake. However, he cannot detect all fakes. In fact the probability that a fake passes Steve’s sniff test is 1/2. Leroy detects fakes in the same way as Steve. Half of the fakes fail his sniff test and half of them pass his sniff test. Genuine paintings are sure to pass Leroy’s sniff test. For any fake, the probability that Steve recognizes it as a fake is independent of the probability that Leroy recognizes it as a fake. The auction house posts a price for each painting. Potential buyers can submit a written offer to buy at the posted price on the day of the sale. If more than one person offers to buy the painting, the auction house will select one of them at random and sell it to that person at the posted price.
(a) One day, as the auction house is about to close, Steve arrives and discovers that neither Leroy nor any other bidders have appeared. He sniffs a painting, and it passes his test. Given that it has passed his test, what is the probability that it is a good painting? (Since fakes are much more common than good paintings, the number of fakes that pass Steve’s test will exceed the number of genuine antiques that pass his test.) _______. Steve realizes that he can buy the painting for the posted price if he wants it. What is the highest posted price at which he would be willing to buy the painting? _______.
(b) On another day, Steve and Leroy see each other at the auction, sniffing all of the paintings. No other customers have appeared at the auction house. In deciding how much to bid for a painting that passes his sniff test, Steve considers the following: If a painting is selected at random and sniffed by both Steve and Leroy, there are five possible outcomes. Fill in the blanks for the probability of each.
Genuine and passes both dealers’ tests. Probability: _______.
Fake and passes both dealers’ tests. Probability: _______.
Fake and passes Steve’s test but fails Leroy’s. Probability: _______.
Fake and passes Leroy’s test but fails Steve’s. Probability: _______.
Fake and fails both dealers’ tests. Probability: _______.
(c) On the day when Steve and Leroy are the only customers, the auction house sets a reserve price of $300. Suppose that Steve believes that Leroy will offer to buy any painting that passes his sniff test. Recall that if Steve and Leroy both bid on a painting, the probability that Steve gets it is only 1/2. If Steve decides to bid on every painting that passes his own sniff test, what is the probability that a randomly selected painting is genuine and that Steve is able to buy it? _______. What is the probability that a randomly selected painting is a fake and that Steve will bid on it and get it? .3. If Steve offers to pay $300 for every painting that passes his sniff test, will his expected profit be positive or negative? _______. Suppose that Steve knows that Leroy is willing to pay the reserve price for any painting that passes Leroy’s sniff test. What is the highest reserve price that Steve should be willing to pay for a painting that passes his own sniff test? _______.