Randomness of the tossed coin

The Quick Study “Probability, physics, and the coin toss” (Physics Today, July 2011, page 66 ) by L. Mahadevan and Ee Hou Yong discusses the dynamics of a fair, thick coin. In particular, the authors try to determine how thick a coin should be to have a one-in-three chance of landing on edge. The authors do not consider coin bounces but note that adding bounces leads to “more physical realism and fun.” Their main result—illustrated in the boundary plot, figure 2c, which presents initial conditions leading respectively to heads, tails, and edges—is stated as follows: “As ω [angular velocity] and u/g [vertical speed scaled by the gravitational acceleration g] become large, any disk representing a probability distribution of initial conditions is tiled finely and equally, now by regions associated with heads, tails, and sides.” That statement is correct.

Note, however, that the boundaries separating the regions of heads, tails, and sides are smooth. Thus if one takes finite initial values for the angular and vertical velocities and takes care to avoid the boundaries, then a small enough initial error will yield a predictable, fixed outcome for the coin toss. In that scenario, bounces are essential for randomness in the coin toss (see reference , which considers an infinitesimally thin coin). As the number of bounces increases, the boundaries of the heads and tails regions become more complex; in the limit of an infinite number of bounces, they become fractal (see reference , figures 3 and 4). In that limit only, it is impossible to predict the outcome of the coin toss no matter how well the initial conditions are determined. We obtained similar results for the dynamics of dice, ² which can be viewed as a generalization of the thick coin considered in the Quick Study.