Will every bin get at least one ball?

Question

You throw 5 balls independently into 3 bins, each ball landing in a uniformly random bin. What is the probability that every one of the three bins ends up with at least one ball?

Accepted Answer

There are 3^5 = 243 equally likely ways to assign the 5 balls. We want the fraction where all three bins are non-empty. "All bins occupied" is hard to count head-on, but its opposite, "at least one bin empty," splits into overlapping cases we can handle with inclusion-exclusion. Let Ai be the event that bin i is empty. - Subtract arrangements missing a chosen bin: there are 3 bins to pick, and if one bin is banned each ball has 2 choices, giving 3 \cdot 2^5 = 96. - Add back arrangements missing two chosen bins (subtracted twice): there are 3 such pairs, and all balls must go to the one remain…

Will every bin get at least one ball?

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