A Short Note on the Average Maximal Number of Balls in a Bin

Abstract

We analyze the asymptotic behavior of the average maximal number of balls in a bin obtained by throwing uniformly at random $r$ balls without replacement into $n$ bins, $T$ times. Writing the expected maximum as $\frac{r}{n}T+ C_{n,r}\sqrt{T} + o(\sqrt{T})$, a recent preprint of Behrouzi-Far and Zeilberger asks for an explicit expression for $C_{n,r}$ in terms of $n,r$ and $\pi$. In this short note, we find an expression for $C_{n,r}$ in terms of $n, r$ and the expected maximum of $n$ independent standard Gaussians. This provides asymptotics for large $n$ as well as closed forms for small $n$---e.g. $C_{4,2} = \frac{3}{2 \pi^{3/2}} \arccos(-1/3)$---and shows that computing a closed form for $C_{n,r}$ is precisely as hard as the difficult question of finding the expected maximum of $n$ independent standard Gaussians.