Je-ok claims he brought a trick coin from a casino. In particular, he claims that the coin is more likey to come up heads than tails. To validate his claim, he flips the coin 10 times, and it comes up heads 7 times.

(a) Assume we start with a uniform prior for the probability p that the coin lands heads. What is the posterior distribution? Plot the density of the posterior and compute the posterior probability that p > 0.5.

(b) Stephen thinks trick coin sellers are the real tricksters and Je-ok has probably been duped. He also believes that all trick coins only come up heads sixty percent of the time. Thus he has the prior that p = 0.5 with probability 0.9, and p = 0.6 with probability 0.1. Give Stephen’s posterior distribution using this prior and again compute the posterior probability that p > 0.5.