Classical Origns of Frequency Probabilities

A classical open problem in probability theory concerns the so called three sided dice: suppose a cylinder with diameter $d$ and height $h$, what should be the ratio $h/d$ so that the frequency to obtain a face is $1/3$ (where falling on the cylinder lateral side counts as a face). In a more general situation, we can ask for what is the frequency $P(S|d,h,H,\theta ,\varepsilon )$ for obtaining a fall on the lateral side $S$ given the cylinder diameter $d$ and tallness $h$, the height $H$ of its center of mass above a table at the moment of launching, the initial angle $\theta $ of the cylinder axis with the horizontal and the elastic coefficient of restitution $\varepsilon $, which depends on the materials of the table and the cylinder. We do not consider here initial conditions with translational or rotational velocities. We made experimental measures for $P(S|d,h,H,\varepsilon )$ varying $h/d$ with $H$ large (so that influences of initial conditions vanishes) and $\varepsilon $ fixed. We also model numerically the system as a two dimensional ``cylinder" of height $h$ and $d$ composed by four masses linked by springs. Given initial conditions, the numerical result is deterministic. However, for $H > 30 cm$, the toss outcome depends strongly on initial conditions, so that we must average over a cell $\Delta H.\Delta \theta $ of initial conditions under control of the experimenter. This average furnishes a frequency $P(S|d,h,H,\varepsilon )$ to be compared to the experimental results. By using the unknown $\varepsilon $ as a free parameter, we obtain a very good agreement between the full three-dimensional experiment and the two-dimensional simulation. The experimental data can also be fitted by a recently proposed ``Gibbs curve".