Coats handed back at random: how many match?

Question

A cloakroom checks coats from 8 guests, but the claim tickets get scrambled. At closing time the attendant simply hands the 8 coats back in a completely random order, one coat per guest.

On average, how many of the 8 guests walk away with their own coat?

Accepted Answer

The number of correct returns is hard to track directly, because whether one guest is right nudges the odds for everyone else. Linearity of expectation lets us ignore all of that and just add up per-guest chances. Pick one guest. In a fully random ordering, that guest's coat is equally likely to reach any of the 8 people, so the chance it comes back to its owner is 	frac18. Define an indicator Xi = 1 if guest i gets the right coat and 0 otherwise. Then E[Xi] = 	frac18. The total number of matches is X = X1 + X2 + \cdots + X8. By linearity, E[X] = 8 	imes \frac18 = 1. Notice the answer does…

Coats handed back at random: how many match?

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