A hat contains 100 coins, where at least 99 are fair, but there may be one that is double-headed (always landing Heads); if there is no such coin, then all 100 are fair.
Let D be the event that there is such a coin, and suppose that P (D) = 1/2. A coin is chosen uniformly at random. The chosen coin is flipped 7 times, and it lands Heads all 7 times.
(a) Given this information, what is the probability that one of the coins is double-headed?
(b) Given this information, what is the probability that the chosen coin is double-headed?