Alice has two coins. The probability of Heads for the rst coin is 1=3, and the probability of Heads for the second is 2=3. Other than this di erence, the coins are indistinguishable. Alice chooses one of the coins at random, such that the rst coin has probability p of being selected, and then sends it to Bob. Assume that 0 < p < 1 and its value is known to Bob. Bob tries to guess which of the two coins he received by tossing it 3 times in a row and observing the outcome. Assume that for any particular coin, all tosses of that coin are independent.

Given that Bob observed k Heads out of the 3 tosses (where k 0,1,2,3), what is the conditional probability that he received the first Coin?