# Is The Probability of Tossing a Coin Really 50–50%? Part 2: Dynamic Model with Rebounds

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Model with Rebounds

#### 2.1. General Model and Equations

- -
- the coin is fair, i.e., the coin is a homogeneous flat circular cylinder of mass $m$ and with thickness $h$ and diameter $d$;
- -
- the coin is thrown manually from an initial height $H$ with a velocity ${v}_{0}$ under an angle $\beta $ on the horizontal, and an initial angular velocity ${\omega}_{0}$; for a manual throw, the minimum and maximum possible values are considered to be:

- -
- the coin rotation axis is horizontal and passes through the coin center of mass at all times during the fall, until impacting the landing surface;
- -
- the coin angular velocity after the impact is along an undefined instantaneous axis of rotation that stays horizontal at all times;
- -
- the atmosphere is windless, without any disturbance and the air friction is negligible;
- -
- the landing surface is a perfectly horizontal plane, with a solid and immovable surface.

- -
- ${v}_{0}$ the norm of the velocity vector before impact;
- -
- ${u}_{\mathrm{x}}$ and ${u}_{\mathrm{z}}$ the components of the velocity vector after impact;
- -
- $\beta $ the angle of the initial velocity vector with the horizontal (see Figure 1);
- -
- T and N are the components of the impulse vector, respectively, along and normal to the surface at the point of contact;
- -
- $\rho =\frac{\sqrt{{h}^{2}+{d}^{2}}}{2}$ is the distance from the impact point to the coin center of mass (see Figure 2);
- -
- ${I}_{\mathrm{b}}$ and ${I}_{\mathrm{a}}$ are the moments of inertia of $m$ at the coin center of mass with respect to the instantaneous axis of rotation before and immediately after impact;
- -
- $\pm {\omega}_{0}$ and $\pm {\omega}_{a}$ are the coin angular velocity before and after impact, + (respectively, -) for a coin counterclockwise (respectively, clockwise) rotation;

#### 2.2. The Case of Rebounds

- -
- between the first and second impacts, the coin attains a maximum height:

- -
- the rotation velocities and moments of inertia of the coin, respectively, before the second impact and after the first impact are the same, yielding:

- -
- the vertical components of the coin velocity before and after the second impact are, respectively:

- -
- the coin and landing surface are elastic and partially rough bodies, such as the normal and tangential impulses read:

- -
- both denominator and numerator are positive if

- -
- both denominator and numerator are negative if:

- -
- if $\mu <\frac{h}{d}$, then (29) includes (27) and ${\omega}_{a1}$ must be positive, i.e., a coin counterclockwise rotation after first impact;
- -
- if $\mu =\frac{h}{d}$, then (30) is the limiting case of (26) and (27) and ${\omega}_{a1}$ is positive, i.e., a coin counterclockwise rotation after first impact;
- -
- if $\frac{h}{d}<\mu <\frac{2hd}{{d}^{2}-{h}^{2}}$, then (31) includes (27) and ${\omega}_{a1}$ is positive, i.e., a coin counterclockwise rotation after first impact;
- -
- if $\mu >\frac{2hd}{{d}^{2}-{h}^{2}}$, then (31) includes (28) and ${\omega}_{a1}$ can be either positive or negative, i.e., a coin counterclockwise rotation or clockwise rotation after first impact.

^{−3}and 1.475 × 10

^{−2}, while ${\kappa}_{1}$ = 2.955 × 10

^{−3}under the initial hypothesis.

^{−3}m and ${\kappa}_{2max}$ = 1.475 × 10

^{−2}m, yielding, respectively, for $H=d$ and for $H=2$ m:

^{−3}m, a series of coin tosses having initial conditions in the ranges (1), one finds, successively:

^{−2}, ${P}_{edge21max}$= 2.225 × 10

^{−2}, ${P}_{edge2max}$= 4.880 × 10

^{−4}

^{−2}m, one has similarly:

^{−2}, ${P}_{edge21max}$ = 2.225 × 10

^{−2}, ${P}_{edge2max}$= 4.868 × 10

^{−4}

^{−3}m:

^{−3}, ${P}_{edge21}$ = 2.224 × 10

^{−2}and ${P}_{edge2}$ = 1.729 × 10

^{−4}

^{−2}m:

^{−3}, ${P}_{edge21}$ = 2.224 × 10

^{−2}and ${P}_{edge2}$ = 1.726 × 10

^{−4}

## 3. Conclusions

- -
- there is a non-nil probability that a falling coin will not end up on one of its sides but on its edge, with decreasing probabilities for the models describing reality from closer;
- -
- probabilities calculated are independent of the coin mass but strongly depend on the coin’s vertical velocity before impact, on the initial height $H$ and on the initial angle $\beta $ of the throw;
- -
- increasing the initial height decreases the probability that the coin will end on its edge, while increasing the initial rotation will increase this probability;
- -
- depending on surface characteristics, tossing the coin vertically decreases the probability of the coin ending on its edge;
- -
- friction is of paramount importance: if the friction coefficient $\mu $ is increased above a certain value depending on surface conditions, the coin can no longer stop on its edge and will inevitably fall on one side.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Pletser, V. Is the probability of tossing a coin really 50-50%? Part 1: Static Model and Dynamic Models without rebounds. Foundations
**2022**, 2, 547–560. [Google Scholar] [CrossRef] - MacMillan, W.D. Dynamics of Rigid Bodies; Dover Publications: New York, NY, USA, 1960; pp. 301–303. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pletser, V.
Is The Probability of Tossing a Coin Really 50–50%? Part 2: Dynamic Model with Rebounds. *Foundations* **2022**, *2*, 581-589.
https://doi.org/10.3390/foundations2030039

**AMA Style**

Pletser V.
Is The Probability of Tossing a Coin Really 50–50%? Part 2: Dynamic Model with Rebounds. *Foundations*. 2022; 2(3):581-589.
https://doi.org/10.3390/foundations2030039

**Chicago/Turabian Style**

Pletser, Vladimir.
2022. "Is The Probability of Tossing a Coin Really 50–50%? Part 2: Dynamic Model with Rebounds" *Foundations* 2, no. 3: 581-589.
https://doi.org/10.3390/foundations2030039