1 Suppose you are given a bag containing n unbiased 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 heads on both sides.

a. Suppose you reach into the bag, pick out a coin at random, flip it, and get a head. What is the (conditional) probability that the coin you chose is the fake coin?

b. Suppose you continue flipping the coin for a total of k times after picking it and see k heads. Now what is the conditional probability that you picked the fake coin?

c. Suppose you wanted to decide whether the chosen coin was fake by flipping it k times.
The decision procedure returns fake if all k flips come up heads; otherwise it returns normal . What is the (unconditional) probability that this procedure makes an error?