Suppose we have four identical-looking coins. Three are fair, but one is biased, with a probability of coming up heads of 3/5. We select one of the four coins at random.
1. If we flip the selected coin twice, and it comes up heads both times, what is the probability that our coin is the biased one?
2. If we flip the selected coin three times, and it comes up heads all three times, what, then, is the probability that our coin is the biased one?
3. Generalize: We have m fair coins and one identical-looking biased coin with probability p of getting heads. If we select one coin at random, and obtain k heads in n flips, what is the probablility P(m,p,n,k) that we have the biased coin?