Will one particular bin stay empty?

Question

You toss 6 balls, one after another, into 3 bins. Each ball independently lands in any of the 3 bins with equal probability. What is the probability that bin number 1 ends up empty?

Accepted Answer

Focus only on bin 1. Bin 1 stays empty exactly when every ball misses it. A single ball misses bin 1 with probability \frac{2}{3}, because 2 of the 3 bins are "not bin 1". The 6 balls are independent, so all six miss with probability \left(\frac{2}{3}\right)^6 = \frac{64}{729} \approx 0.0878. So bin 1 is empty only about 8.8% of the time. The other bins have the same chance individually, though of course they cannot all be empty at once. Takeaway. For m balls dropped uniformly into n bins, a specific bin is empty with probability \left(\frac{n-1}{n}\right)^m, since each ball independently avo…

Will one particular bin stay empty?

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