Probability and dynamics in the toss of a non-bouncing thick coin

TL;DR

This paper models the physics of tossing a thick cylindrical coin, deriving probabilities for landing on face, side, or tail based on its dimensions and initial conditions, validated by experiments.

Contribution

It introduces a theoretical framework for predicting the landing probabilities of thick coins, incorporating rigid body dynamics and validating with experimental data.

Findings

Derived a simple expression for aspect ratio related to landing probabilities

Validated theoretical predictions with experimental coin tosses

Provided insights into the dynamics of non-bouncing thick coins

Abstract

When a thick cylindrical coin is tossed in the air and lands without bouncing on an inelastic substrate, it ends up on its face or its side. We account for the rigid body dynamics of spin and precession and calculate the probability distribution of heads, tails, and sides for a thick coin as a function of its dimensions and the distribution of its initial conditions. Our theory yields a simple expression for the aspect ratio of homogeneous coins with a prescribed frequency of heads/tails compared to sides, which we validate by tossing experiments using coins of different aspect ratios.