An elevator nobody rides

Question

In a lobby, 8 people each independently choose one of 7 elevators, uniformly at random. What is the probability that elevator A carries none of the eight people?

Accepted Answer

Look only at elevator A. It stays empty when all 8 riders pick one of the other elevators. A single rider avoids elevator A with probability \frac{6}{7} (6 of the 7 elevators are "not A"). Since choices are independent, \left(\frac{6}{7}\right)^8 = \frac{6^8}{7^8} = \frac{1{,}679{,}616}{5{,}764{,}801} \approx 0.2914. So about 29% of the time a specific elevator gets skipped entirely, even with eight riders. Takeaway. The chance a particular bin is empty is \left(\frac{n-1}{n}\right)^m; when balls and bins are close in number, this can stay comfortably above a quarter.

An elevator nobody rides

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