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

Area of Science:

• Physics
• Statistical Mechanics
• Probability Theory

Background:

• Traditional coin tosses assume two outcomes (heads/tails) based on equiprobability.
• The possibility of a coin landing on its edge is often disregarded.
• Previous research on three-outcome coin tosses relied solely on computational methods.

Purpose of the Study:

• To derive an exact analytical solution for the probabilities of all three coin toss outcomes (heads, tails, edge).
• To investigate coin tosses with arbitrarily inelastic bouncing.
• To propose optimal geometries for fair, three-sided dice.

Main Methods:

• Combined classical collision equations with a statistical-mechanics approach.
• Developed an analytical solution for bouncing object dynamics.
• Validated theoretical predictions against simulations and experimental data.

Main Results:

• An exact analytical solution for the three outcome probabilities of a bouncing coin toss was derived.
• Theoretical predictions showed moderate discrepancies in the highly inelastic regime.
• The study identified optimal geometries for a fair cylindrical three-sided die.

Conclusions:

• The developed analytical model accurately predicts coin toss outcomes, including the edge landing.
• This work provides a significant advancement beyond previous approximate and computational models.
• The findings have implications for probability, physics, and the design of fair dice.