Exactly one of four diners gets their own hat

Question

Four diners receive their four hats back in a completely random order, so all 4! = 24 assignments are equally likely. What is the probability that exactly one diner gets their own hat?

Accepted Answer

Exactly one match means one diner is matched and the other three are all mismatched. Count in two steps: 1. Choose the single matched diner: \binom{4}{1} = 4 ways. 2. Derange the remaining three hats so nobody else matches: D3 = 2 ways. So there are 4 	imes 2 = 8 favorable arrangements, and P(	ext{exactly one match}) = \frac{4 \cdot D3}{4!} = \frac{8}{24} = \frac{1}{3} \approx 0.3333. Takeaway. Exactly one fixed point uses the same recipe: choose the matcher, then derange everyone else.

Exactly one of four diners gets their own hat

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