George has approached Betsy for a date and been rejected on multiple occasions.

Question

George has approached Betsy for a date and been rejected on multiple occasions. He is trying to determine whether Betsy really likes him while playing hard to get or whether she genuinely does not like him. Knowing that probability tree he learned is a useful tool in this case, George first defines the following events: L = Betsy likes George, N = Betsy does not like George. a = Betsy accepts George, r = Betsy rejects George. He is eager to find out P(L/r), the probability that Betsy likes George given her rejection. Using his subjective judgment, he estimates: P(r|L) = the probability of rejection given Betsy likes George = 0.05 George recognizes that this conditional probability must be quite low because Betsy is known to be a very honest and straightforward person. If Betsy actually likes George, he would expect an honest admission from her, rather than repeated rejections. However, women can be funny in showing their affection, so P(r|L) is still greater than 0. It's very low, but leaves a little room for error in case his perception of Betsy has been mistaken. It is obvious that P(a|L) = 0.95 (Why?) If Betsy does not like George, then it is almost certain she would reject him. Considering Betsy's honest manner, he assigns: P (r|N) = the probability of rejection given Betsy does not like George = 0.99 It is obvious that P(a|n) = 0.01 (Why?) However, George is popular among women in general and he estimates: P(L) = the probability that Betsy likes George = 0.9 This prior probability must be high since, in determining P(L), the evidence such as the rejections should not count. In other words, P(L) is the probability that Betsy likes George prior to rejections (or the probability that a woman likes George in general). So, he can err on the side of his prominent features such as good looks, size and so on. It is obvious that P(N) = 0.1 (Why?) (George and Betsy) Find the joint probabilities below. (a) Probability that Betsy accepts George and Betsy likes George [Answer format: three decimal places] (b) Probability that Betsy rejects George and Betsy likes George [Answer format: three decimal places] (c) Probability that Betsy accepts George and Betsy does not like George [Answer format: three decimal places.] (d) Probability that Betsy rejects George and Betsy does not like George [Answer format: three decimal places] Write your answer(s) as 0.123, 0.456, 0.789, 0.234

Explanation / Answer

(a) P(L,a) = P(L)*P(a/L) = 0.9*.95 = 0.855

(b) We must have P(L,a) + P(L,r) = P(L) since Betsy either accepts George or rejects him, there can't be any other possibility. or P(L,r) = P(L) - P(L,a) = 0.9 - 0.855 (from the previous answer) = 0.045. Another way to find this is P(L,r) = P(L)*P(r/L) = .9*.05 = 0.045. The same answer!

(c) P(N,a) = P(N)*P(a/N) = [1-P(L)]*P(a/N) = (1-.9)*.01 = 0.001.

(d) P(N,r) = P(N)*P(r/N) = [1-P(L)]*P(r/N) = (1-.9)*.99 = 0.099.

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