About a Ball Removal Process on Bins
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Implication
Theoretical analysis demonstrates optimal ball clearance in randomized bin systems, confirming that balanced initial distributions minimize surviving elements.
Key Points
- Determine whether an initial balanced distribution of balls into bins minimizes the expected number of remaining balls when randomly clearing nonempty bins until only one holds balls.
- Modeled a discrete stochastic process distributing n balls into k bins with rounds of uniform random removal from nonempty bins until a single nonempty bin remains.
- Applied a probabilistic coupling argument to compare arbitrary starting configurations against balanced ball distributions, with extensions to nonuniform bin selection.
- Positively resolved Will Ma's conjecture by mathematically proving that maximally balanced initial ball assignments minimize the expected number of leftover balls.
- Demonstrated the analytic bounds and behavioral dynamics that occur when nonempty bins are selected under nonuniform choice probabilities.
Cite This Study
Correa et al. (2026) studied this question.