Montmort’s Theorem

Suppose n students are sitting at n desks in a classroom. They’re asked to stand, mill around at random, and then sit again. What is the probability that at least one student will find herself in her original seat?

Intuition says that the probability ought to drop as the number of students increases, but in fact it remains about the same:

montmort table

In fact, Pierre Rémond de Montmort showed in 1708 that it’s

montmort theorem

… which approaches 1 – 1/e, or about 0.63212. Whether there are 10 students or 10,000, the chance that at least one student returns to her own seat is about 2/3.