Problem: There are M balls and N boxes. Each ball is randomly placed into a box. What is the expected number of boxes that have at least one ball inside?
Solution: We can create a series of identification variables, Ii, which denotes for the i th box, if there is at least one ball inside. If yes, Ii = 1, otherwise Ii = 0. Then our problem is to calculate the expected value to I, where I = I1 + I2 + ... + IN. There is symmetric here, so E(I) = N*E(Ii) = N*(1-(N-1/N)^M)
Related: How many boxes you need to have in order to collect all the toys?
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