Consider three boxes. Box 1 contains two gold coins, box 2 contains two silver coins, and box 3 contains one gold and one silver coin. A box is chosen at random and then the coins in that box are chosen one at a time. Suppose that the first coin is gold. What is the probability that the second coin is also gold?
On the one hand, since the box chosen must be 1 or 3, it would seem that the next coin is either gold or silver with probability 1/2 for each.
On the other hand, it is more likely that the first gold coin comes from box 1 than box 3, leading to the probability 2/3 .