Empty stockings on the mantel

Question

Twelve identical gifts are placed one at a time into 8 stockings, each gift going into a uniformly random stocking, independently. On average, how many stockings will be empty at the end?

Accepted Answer

Use linearity of expectation: the expected number of empty stockings is the sum of the probabilities that each individual stocking is empty. A given stocking gets missed by one gift with probability \frac{7}{8}, so it stays empty through all 12 gifts with probability \left(\frac{7}{8}ight)^{12} \approx 0.2014. There are 8 stockings, each with the same chance, so E[	ext{empty}] = 8 	imes \left(\frac{7}{8}ight)^{12} \approx 8 	imes 0.2014 = 1.611. Takeaway. Expected empty bins is n\left(\frac{n-1}{n}ight)^m; here about 1.6 of the 8 stockings go empty on average.

Empty stockings on the mantel

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