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.

翻译：假设 $2n$ 个球独立且均匀随机地投入 $n$ 个箱子中。我们考虑每个箱子至少有一个球的小概率事件 $E$，证明 $\Pr[E] = \Theta(b^n)$，其中 $b \approx 0.836$。注意，由于相关性，$b$ 并非任意单个箱子至少有一个球的简单概率。更一般地，我们考虑将 $\alpha n$ 个球投入 $n$ 个箱子后每个箱子至少包含 $d$ 个球的事件。