A Coin, First Throw Get Head, Bet the Second Thow

Problem: A coin (you don't know it is a fair one or biased one), in the first throw, you got a "head". What will you bet for the second throw and what is the probability to win?

Solution: This is a conditional probability problem. Assume the probability to get head is p. f(p) is a uniform distribution. P(H2/H1) = P(H2H1)/P(H1) = int_{p^2*f(p)*dp} / int_{p*f(p)*dp} = (1/3) / (1/2) = 2/3.