When Raymond Smullyan was teaching probability at Princeton, he told one class about the birthday paradox — the fact that if there are 23 people in a room, the chances are greater than 50 percent that at least two of them share a birthday. There were only 19 students in the classroom, so he said that the chance that two of them shared a birthday was quite small. One boy said, “I’ll bet you a quarter that two of us here have the same birthday.” Smullyan thought about that for a moment and said, “Oh, of course! You know the...