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midtermsolutions

midtermsolutions - Midterm Exam Solutions Ap.Sc 116 Problem...

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Midterm Exam Solutions Ap.Sc. 116 Problem 1. Suppose that a box contains five coins, and that for each coin there is a different probability that a head will be obtained when the coin is tossed. Let p i denote the probability of a head when the i th coin is tossed ( i = 1 , 2 , 3 , 4 , 5 ) and suppose that p 1 = 0 , p 2 = 1 / 3 , p 3 = 2 / 3 , p 4 = 5 / 6 , and p 5 = 1 . (a) Suppose that one coin is selected at random from the box and when it is tossed once, a head is obtained. What is the posterior probability that the i th coin was selected ( i = 1 , . . . , 5 )? (b) If the same coin were tossed again, what would be the probability of obtaining another head? (a) Define C i as the event that coin i , and H the event that head shows up, we apply Baye’s rule to obtain Pr ( C i | H ) = Pr ( H | C i ) Pr ( C i ) 5 j =1 Pr ( H | C j ) Pr ( C j ) = 1 5 p i 1 5 5 j =1 p j = p i 5 j =1 p j . Since 5 j =1 p j = 17 / 6. This yields the following values i 1 2 3 4 5 p i P j p j 0 2 17 4 17 5 17 6 27 (b) If coin i was selected, the probability of another head is simply p i .
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