Suppose you are given a bag containing n coins. You are told that n ? 1 of these

Question

Suppose you are given a bag containing n coins. You are told that n ? 1 of these coins are normal, with heads on one side and tails on the other, whereas one coin is a fake, with headson both sides.

(a) Suppose you reach into the bag, pick out a coin uniformly at random, ?ip it, and get

heads. What is the probability that you picked the fake coin?

(b) Suppose you continue ?ipping the coin for a total of k times after picking it and get k

heads. Now what is the probability that you picked the fake coin?

Explanation / Answer

The conditional probability of A given B is defined as the quotient of the joint probability of A and B, and the marginal probability of B

P(A|B) = P(A n B)/P(B)

(a) probability of random coin being head P(B) = (n-1)/n * (1/2) + 1/n*1 = (3n - 1)/2n.

probability that the random coin flipped head is a fake coin = P(A n B)= 1/n * 1

Hence ,

conditional probability that the coin you chose is the fake coin after you pick out a coin at random, flip it, and get a head =
P(A|B) = P(A n B) / P(B) = (1/n) / [(3n-1) / 2n ] = 2/(3n-1).

(b)probability that you picked the fake coin and you contine flipping the coin for a total of k times after picking it and see k heads = P(A n B) = 1/n *1 =1/n

probability that after flipping a coin for k times and see k heads = P(B) = [(n-1)/n ]*(1/2)^k + 1/n * 1 = (n + 2^k -1)/(n*2^k)

conditional probability that you picked the fake coin = (1/n) / [(n + 2^k -1)/(n*2^k)]
= 2^k/ (n + 2^k -1)

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