Question

Tom has three coins. Two are fair and one is unfair coin weighted so that heads is three times as likely as tails. He selects one of the coins at random and flips it. What is the probability it comes up heads? If it does come up heads, what is the probability it was the unfair coin?

Answer 1

P(Fair coin) = 2/3

P(Unfair coin) = 1/3

P(Heads | Fair coin) = 1/2

P(Heads | Unfair coin) = 3/4

Hence,

P(Heads) = P(Heads | Fair coin) * P(Fair coin) + P(Heads | Unfair coin) * P(Unfair coin) = (1/2)*(2/3) + (3/4)*(1/3) = 7/12

P(Unfair coin | Heads) = ((3/4)*(1/3)) / (7/12) = 3/7