The wily Sheriff of Nottingham wants to catch the outlaw Robin Hood and his Merry Men. They are excellent archers, so he decides to hold an archery contest to figure out who they are.

Each contestant will shoot several arrows. On each shot, the Merry Men hit the bullseye with probability 9/10. There will also be villagers at the contest, who hit the bullseye with probability 1/10 on each shot. Suppose that, of the contestants who show up, 1/41/4 are Merry Men and 3/4 are Villagers.

(a) A contestant is chosen randomly. Given that he hits the bullseye on the first shot, what is the probability that he is a Merry Man?

(b) A contestant is chosen randomly. What is the probability that he will miss the bulls eye on his second shot given that he misses it on the first shot?

Assume the conditional probability of missing the second shot only depends on whether it is a Merry Man or Villager, and is not directly affected by whether the first shot was missed.