Folders that receive no email

Question

A filing rule sends each of 15 incoming emails into one of 12 folders, choosing the folder uniformly at random and independently for each email. On average, how many of the 12 folders end up empty?

Accepted Answer

By linearity of expectation, add up the per-folder empty probabilities. Each folder has the same chance, so the answer is 12 times that chance. A single email misses a given folder with probability \frac{11}{12}, so the folder stays empty through all 15 emails with probability \left(\frac{11}{12}ight)^{15} \approx 0.2711. Thus E[	ext{empty}] = 12 	imes \left(\frac{11}{12}ight)^{15} \approx 12 	imes 0.2711 = 3.254. With more folders than emails, several folders are expected to stay empty. Takeaway. Expected empty bins is n\left(\frac{n-1}{n}ight)^m; when m < n a sizeable fraction of b…

Folders that receive no email

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