Lonely passengers: a short proof

Abstract

A fixed number of passengers independently board one of several buses uniformly at random. The lonely passenger problem is to prove that the probability of at least one passenger being the only one in their bus is increasing in the number of buses. It was solved in a strong form by Imre P\'eter T\'oth, who proved stochastic dominance of the number of such passengers as the number of buses increases, but observed that, surprisingly, no short proof was known ``despite the efforts of several experts''. We give a very short proof of the weaker result. The proof of the strong form, using the same idea, is more involved but still relatively short.