There are 5 identical looking coins in your pocket. Two of the coins are fair while the other three are biased so that about 75% of the time they come up heads when flipped. You put your hand in your pocket and select a coin at random. You flip the coin 10 times and observe that heads appear 6 times.
(a) What is the (prior) probability that the selected coin is fair, i.e., the unconditional probability that the selected coin is fair?
(b) What is the (posterior) probability that the selected coin is fair, i.e. the conditional probability that the selected coin is fair given that heads appear 6 times out of 10 flips.