Lonely passenger problem: the more buses there are, the more lonely passengers there will be

Abstract

Empty buses are standing at a bus station. $n$ passengers arrive, and they each board a bus completely at random (meaning that they choose uniformly and independently). Then all buses depart. We show that the more buses there are, the more likely it is that someone (i.e. at least one passenger) travels alone (while $n$ is fixed). More generally, we show that the number of lonely passengers increases with the number of buses, in the sense of stochastic dominance. This problem turned out to be surprisingly difficult, with no short solution known to the author so far, despite the efforts of many experts. Some of the results can also be formulated as properties of Stirling numbers of the second kind.