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.
1. 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?
2. 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?
3. 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?