How many boxes stay empty?

Question

You throw 4 balls, one at a time, into 3 boxes. Each ball lands in one of the 3 boxes independently and uniformly at random. What is the expected number of boxes that remain empty at the end?

Accepted Answer

Trying to track the whole distribution of ball counts is messy. Linearity of expectation makes it easy: add up the chance each individual box is empty. A fixed box stays empty only if all 4 balls avoid it. Each ball misses that box with probability 	frac23, so P(	ext{a given box is empty}) = \left(	frac23ight)^4 = \frac{16}{81}. Let X be the number of empty boxes. Writing X as a sum of three indicator variables (one per box) and using linearity: E[X] = 3 	imes \frac{16}{81} = \frac{48}{81} = \frac{16}{27} \approx 0.593. Takeaway. Linearity of expectation sums per-item probabilities even…

How many boxes stay empty?

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