How many bins stay empty on average?

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You toss 10 balls independently into 6 bins, each ball landing in a uniformly random bin. On average, how many of the 6 bins will be empty when you finish?

Accepted Answer

Counting empty bins directly is messy because bins interact. Linearity of expectation avoids that: the average of a sum is the sum of the averages, even when the parts are not independent. Give bin i an indicator Xi that equals 1 if bin i is empty and 0 otherwise. The number of empty bins is X1 + X2 + \cdots + X6, so its average is just the sum of the six individual probabilities. A single bin is empty when all 10 balls miss it. Each ball misses a given bin with probability \frac{5}{6}, so P(	ext{bin } i 	ext{ empty}) = \left(\frac{5}{6}ight)^{10} \approx 0.1615. By symmetry every bin has…

How many bins stay empty on average?

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