Dynamics and Probability in the Toss of a Coin with Symmetric Inhomogeneous Density

TL;DR

This paper analyzes the dynamics and limiting probability of heads in a symmetric inhomogeneous coin toss, extending previous formulas to account for inhomogeneity and initial conditions.

Contribution

It introduces a new probability formula for inhomogeneous coins, expanding upon existing models for homogeneous coins.

Findings

Derived the dynamic behavior of the coin's normal vector.

Calculated the limiting probability of heads based on initial parameters.

Extended classical probability formulas to inhomogeneous cases.

Abstract

Under investigation in this paper is the dynamics and probability of heads in the toss of a coin with symmetric inhomogeneous density. Such coins are assumed to have diagonal inertia matrix. The rotational motion of the coin is determined by the initial angular momentum and initial position of the coin. We described the dynamic behavior of the unit normal vector and calculated the limiting probability of heads as time goes to infinity with respect to the fixed initial parameters. Our probability formula extends the formula for homogeneous coins by Keller and Diaconis et al.