Valet returns cars at random: how many luxury owners win?

Question

A valet stand parks 15 cars but the key tags fall off. At the end of the evening the valet returns the 15 cars to the 15 owners in a completely random one-to-one matching. Exactly 3 of the 15 cars are luxury cars.

On average, how many of the 3 luxury owners get their own car back?

Accepted Answer

We only need to track the 3 luxury owners, so use one indicator per luxury car. Consider one luxury owner. In a random matching, their car is equally likely to reach any of the 15 owners, so the chance it returns to its rightful owner is 	frac{1}{15}. Let Xi = 1 if luxury owner i gets their own car. Then E[Xi] = 	frac{1}{15}. Summing over the 3 luxury owners, X = X1 + X2 + X3, and by linearity E[X] = 3 	imes \frac{1}{15} = \frac{3}{15} = \frac15 = 0.2. All 15 owners together still average 15 	imes 	frac{1}{15} = 1 correct return; the luxury owners make up one fifth of that expected match…

Valet returns cars at random: how many luxury owners win?

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