How many people get their own hat back?

Question

At a coat check, n people hand in identical-looking hats. At the end of the night the attendant, having lost the tickets, returns the n hats to the n people in a uniformly random order (a random permutation). What is the probability a specific person, say person 7, gets their own hat back? And what is the expected number of people who get their own hat?

Accepted Answer

One person's chance. By symmetry, person 7's hat is equally likely to go to any of the n people, and equivalently person 7 is equally likely to receive any of the n hats. Exactly one of those hats is theirs, so P(	ext{person 7 gets own hat}) = \frac{1}{n}. No conditioning on anyone else is needed; the marginal for one person in a random permutation is uniform. Expected number of matches. Let Ij = 1 if person j gets their own hat, else 0. Each Ij has P(Ij = 1) = 	frac{1}{n} from the argument above. By linearity of expectation (which does not care that the Ij are dependent), E[\text{matches}…

How many people get their own hat back?

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