Untouched boxes after twenty tosses

Question

Twenty balls are thrown independently into 10 boxes, each ball equally likely to land in any box. On average, how many boxes remain empty?

Accepted Answer

By linearity of expectation, the expected number of empty boxes equals ten times the chance that one particular box is empty. Each ball misses a given box with probability \frac{9}{10}, so the box survives all 20 throws empty with probability \left(\frac{9}{10}ight)^{20} \approx 0.1216. Therefore E[	ext{empty}] = 10 	imes \left(\frac{9}{10}ight)^{20} \approx 10 	imes 0.1216 = 1.216. Even with twice as many balls as boxes, roughly one box tends to be missed. Takeaway. Expected empty bins is n\left(\frac{n-1}{n}ight)^m. When m = 2n, this is about n e^{-2} \approx 0.135\,n, matching the…

Untouched boxes after twenty tosses

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