B Suppose that a box contains five coins and that for each c

B. 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 tssed. Let probability of a head when the ith coin is tossed (i=1, . . . , 5), and suppose that p1 = 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?

Solution

a)

Let

P(i) = the probability that the ith coin is selected
P(head) =a head results from the toss

Here,

P(i|head) = P(i and head) / P(head)

Here,

P(head) = P(1) P(head|1) + P(2) P(head|2) + P(3) P(head|3) + P(4) P(head|4) + P(5) P(head|5)

As P(i) = 1/5,

P(head) = (1/5)(0) + (1/5)(1/4) +(1/5)(1/2) + (1/5)(3/4) + (1/5)(1) = 1/2

Also,

P(i and head) = P(i) P(head|i) = (1/5) p_i

Thus,

P(i|head) = (1/5) p_i / (1/2)

P(i|head) = (2/5) p_i [ANSWER]

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B)

If the same coin is tossed, then we already know what coin it is, and its probability.

Hence,

P(head|i) = p_i [ANSWER]