One urn left untouched by ten balls

Question

Ten balls are dropped one by one, each landing in one of 2 urns with equal probability, independently. What is the probability that the left urn ends up empty?

Accepted Answer

The left urn is empty only if all 10 balls choose the right urn. Each ball avoids the left urn with probability \frac{1}{2}, so \left(\frac{1}{2}\right)^{10} = \frac{1}{1024} \approx 0.000977. That is under a tenth of a percent: with only two urns and ten balls, it is nearly certain both urns get used. Takeaway. With just two bins the empty-bin formula \left(\frac{n-1}{n}\right)^m becomes \left(\frac12\right)^m, which decays extremely fast as balls pile up.

One urn left untouched by ten balls

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