Exact edge landing probability for the bouncing coin toss and the three-sided die problem

TL;DR

This paper derives the exact analytical probabilities of a coin landing on its edge, side, or face by combining collision physics with statistical mechanics, and proposes an optimal three-sided die design.

Contribution

It provides the first exact analytical solution for the three-outcome landing probabilities of a partially inelastic coin toss, advancing the physics understanding of such phenomena.

Findings

Validated theoretical predictions with simulations and experiments.

Identified discrepancies in highly inelastic regimes.

Proposed optimal geometry for a fair three-sided die.

Abstract

Have you ever taken a disputed decision by tossing a coin and checking its landing side? This ancestral "heads or tails" practice is still widely used when facing undecided alternatives since it relies on the intuitive fairness of equiprobability. However, it critically disregards an interesting third outcome: the possibility of the coin coming at rest on its edge. Provided this evident yet elusive possibility, previous works have already focused on capturing all three landing probabilities of thick coins, but have only succeeded computationally as an exact and fully grounded analytical solution still evades the physics community. In this work we combine the classical equations of collisions with a statistical-mechanics approach to derive the exact analytical outcome probabilities of the partially inelastic coin toss. We validate the theoretical prediction by comparing it to previously reported simulations and experimental data; we discuss the moderate discrepancies arising at the highly inelastic regime; we describe the differences with previous, approximate models; and we finally propose the optimal geometry for the fair, cylindrical three-sided die based on the current results.