I had this question as a bonus problem on a previous exam and thought it was interesting, but I had no idea how to tackle it.
There are three coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75% of the time. When one of the three coins is selected at random and flipped, it shows heads. What is the probability that it was the first coin?