13. Each coin in a bin has a value attached to it. Each time that a coin with value p is flipped it lands on heads with probability p. When a coin is randomly chosen from the bin, its value is uniformly distributed on (0, 1). Suppose that after the coin is chosen but before it is flipped, you must predict whether it will land heads or tails. You will win 1 if you are correct and will lose I otherwise.

(a) What is your expected gain if you are not told the value of the coin?

(b) Suppose now that you are allowed to inspect the coin before it is flipped, with the result of your inspection being that you learn the value of the coin. As a function of p, the value of the coin, what prediction should you make?

(e) Under the conditions of part (b), what is your expected gain?