This is a combinatorics problem, and I think it involves expected values and conditional probability, but I don't know how to use them:
"A bag contains an infinite number of coins whose probabilities of heads on any given flip are uniformly and continuously distributed between 0 and 1. A coin is drawn at random from this bag. Given that the first flip is a head, determine the probability that the next flip is also a head."
The answer is 2/3, but could someone please explain why?