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

Stefan Walzer

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 $\Pr[E] = Θ(b^n)$ 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 $αn$ balls into $n$ bins results in at least $d$ balls in each bin.

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

Abstract

Assume that balls are thrown independently and uniformly at random into bins. We consider the unlikely event that every bin receives at least one ball, showing that where . Note that, due to correlations, is not simply the probability that any single bin receives at least one ball. More generally, we consider the event that throwing balls into bins results in at least balls in each bin.
Paper Structure (8 sections, 6 theorems, 1 equation, 1 figure, 1 table)

This paper contains 8 sections, 6 theorems, 1 equation, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related Work and Motivation
  3. Simple Considerations
  4. An upper bound.
  5. Proof of Theorem \ref{['thm:main']} (i): The Base of the Exponential
  6. Proof idea.
  7. Proof of Lemma \ref{['lem:prob-N_Z=αn']} using Log-Concavity
  8. Sagemath code and tabulated values

Key Result

Theorem 1

Figures (1)

  • Figure 1: Let $n = 2$, $α = 5$ and $d = 3$. The multinomial distribution $(X₁,X₂)$ automatically satisfies $X₁+X₂ = αn$ (diagonal line). A pair $(Z₁,Z₂)$ of truncated Poisson random variables automatically satisfies $Z₁ ≥ d$ and $Z₂ ≥ d$ (gray). This gives us two perspectives on the outcomes relevant for $E$ (blue), which satisfy both conditions.

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 2: SW:Log-Concavity-Review:2014
  • Corollary 3
  • Lemma 4: BMM:Concentration-LogConcave:2021
  • Lemma 5
  • Lemma 6