The Probability to Hit Every Bin with a Linear Number of Balls

TL;DR

This paper analyzes the probability that, when throwing a linear number of balls into bins, each bin receives at least one ball, revealing that this probability decreases exponentially with the number of bins.

Contribution

It provides a precise asymptotic estimate for the probability of all bins being occupied when balls are thrown uniformly at random.

Findings

Probability of all bins occupied is approximately .836^n.

Probability decreases exponentially with number of bins.

Generalizes to events with at least d balls per bin.

Abstract

Assume that $2n$ balls are thrown independently and uniformly at random into $n$ bins. We consider the unlikely event $E$ that every bin receives at least one ball, showing that $\Pr[E] = \Theta(b^n)$ where $b \approx 0.836$. Note that, due to correlations, $b$ is not simply the probability that any single bin receives at least one ball. More generally, we consider the event that throwing $\alpha n$ balls into $n$ bins results in at least $d$ balls in each bin.