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

Stefan Walzer

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arXiv · 2403.00736

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

Stefan Walzer

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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.

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Stefan Walzer. 2024-03-01. The Probability to Hit Every Bin with a Linear Number of Balls. https://arxiv.org/abs/2403.00736

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