Variance of the number of empty boxes after tossing two balls

Question

Two balls are tossed independently into three boxes, each ball equally likely to land in any box. Let X count how many of the three boxes end up empty. What is the variance of X?

Accepted Answer

Write X as a sum of three counters, Ik = 1 if box k is empty. These counters are linked: if one box is empty, the balls are squeezed into the others, changing their chances of being empty too. Each counter's own variance. A specific box stays empty only if both balls miss it, chance \left(	frac23ight)^2 = 	frac49. So each counter has variance 	frac49\cdot	frac59 = 	frac{20}{81}, and there are three of them. Covariance between two boxes. Two specific boxes are both empty only if both balls land in the one remaining box, chance \left(	frac13ight)^2 = 	frac19. So \mathrm{Cov} = 	frac…

Variance of the number of empty boxes after tossing two balls

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