Secret Santa draws: how many pick themselves?

Question

Twelve coworkers run a Secret Santa. Each person's name goes into a hat, and each person draws one name at random to be their gift recipient. The draw is a completely random one-to-one assignment, and nothing stops someone from drawing their own name.

On average, how many of the 12 people draw their own name?

Accepted Answer

Rather than worry about how one bad draw forces others, count one person at a time using indicators. Fix a single coworker. Their own name is equally likely to be the one they draw as any of the 12 names, so the chance they draw themselves is 	frac{1}{12}. Let Xi = 1 if person i draws their own name. Then E[Xi] = 	frac{1}{12}. The number of self-draws is X = X1 + \cdots + X{12}, and by linearity of expectation E[X] = 12 	imes \frac{1}{12} = 1. The group size cancels out: any random one-to-one assignment produces, on average, exactly one self-match. That is why organizers usually redraw if …

Secret Santa draws: how many pick themselves?

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