Exact face-landing probabilities for bouncing objects: Edge probability in the coin toss and the three-sided die problem

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. Hence, an exact analytical solution for the toss of bouncing objects still remains an open problem due to the strongly nonlinear processes induced at each bounce. In this Letter we combine the classical equations of collisions with a statistical-mechanics approach to derive an exact analytical solution for the outcome probabilities of the toss of a bouncing object, i.e., the coin toss with arbitrarily inelastic bouncing. 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; we propose the optimal geometry for the fair cylindrical three-sided die; and we finally discuss the impact of current results within and beyond the coin toss problem.