Suppose that a box contains one fair coin and one coin with a head on each side. Suppose that a coin is drawn at random from this box and that we begin to flip the coin. In Eqs. (2.3.4) and (2.3.5), we computed the conditional probability that the coin was fair given that the first two flips both produce heads.
a. Suppose that the coin is flipped a third time and another head is obtained. Compute the probability that the coin is fair given that all three flips produced heads.
b. Suppose that the coin is flipped a fourth time and the result is tails. Compute the posterior probability that the coin is fair.