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 pi denote the probability of a head when the ith coin is tossed (i = 1, . . . , 5), and suppose that p = 0, p2 = 1/4, p3 = 1/2, p4 = 3/4, and p5 = 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 ith coin was selected (i = 1, . . . , 5)?
b. If the same coin were tossed again, what would be the probability of obtaining another head?
c. If a tail had been obtained on the first toss of the selected coin and the same coin were tossed again, what would be the probability of obtaining a head on the second toss?