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

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.

💡 Research Summary

The paper investigates the outcome probabilities when a thick cylindrical coin is tossed and lands without bouncing on an inelastic surface. By treating the coin as a rigid body, the authors derive equations of motion that incorporate spin, precession, and the orientation of the rotation axis. Initial conditions—specifically the tilt angle, angular velocity, and axis direction—are modeled as a probability density function. The key geometric parameter is the aspect ratio γ = thickness / (2 × radius). An analytical expression for the critical tilt angle α_c = arctan(2 × thickness / diameter) separates outcomes: if the tilt at impact is smaller than α_c the coin rests on its side, otherwise it lands on one of its faces. Assuming a uniform distribution of initial axis directions and sufficiently large spin, the side‑landing probability P_side reduces to a simple function of γ; the authors show that γ ≈ √2 yields P_side ≈ 1/3, meaning heads, tails, and sides occur with equal likelihood.

To validate the theory, the authors fabricated coins of various γ values from metal and plastic, performed more than 10 000 tosses for each, and recorded the landing face using high‑speed video and automated sensors. The empirical frequencies of heads, tails, and sides matched the theoretical curves across the full range of aspect ratios, with the predicted equality near γ ≈ 1.41 confirmed experimentally. Additional tests varying the spin rate demonstrated that higher spin suppresses precession, thereby increasing the side probability, in line with the model.

The study concludes that the probability distribution of a tossed thick coin can be accurately predicted from its geometry and the statistical properties of its launch conditions. This insight has practical implications for designing tokens or random‑selection devices where a non‑uniform outcome distribution is desired, and it provides a concrete, physics‑based example for teaching probability and rigid‑body dynamics.