The Probability to Hit Every Bin with a Linear Number of Balls
<|reference_start|>The Probability to Hit Every Bin with a Linear Number of Balls: 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.<|reference_end|>
@article{walzer2024the,
    title={The Probability to Hit Every Bin with a Linear Number of Balls},
    author={Stefan Walzer},
    journal={arXiv preprint arXiv:2403.00736},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.00736},
    primaryClass={math.PR cs.DS}
}