# Is the Probability of Tossing a Coin Really 50–50%? Part 1: Static Model and Dynamic Models without Rebounds

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Static Model

## 3. Dynamic Model

#### 3.1. General Model and Equations

- -
- the coin is thrown manually from an initial height $H$ with a velocity ${\mathit{v}}_{\mathbf{0}}$ under an angle $\beta $ on a horizontal surface, and an initial angular velocity ${\mathit{\omega}}_{\mathbf{0}}$; 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 impact is along a (yet) 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, solid and immovable surface;
- -
- in the first approach, one considers that there is no rebound of the coin; the rebound case is addressed in the second part of the paper [3].

- (1)
- both bodies are inelastic and perfectly rough,
- (2)
- both bodies are inelastic and perfectly smooth,
- (3)
- both bodies are elastic and perfectly smooth,
- (4)
- both bodies are elastic and partially rough.

#### 3.2. Inelastic and Perfectly Rough Bodies

- -
- for both numerator and denominator to be positive, the first root must be greater than the second, yielding

- -
- for both numerator and denominator to be negative, the first root must be smaller than the second, yielding

_{,}it yields

^{−3}or approximately 1 throw every 202. For another series of throws with $\beta =\pi /4$, $H=$ 1.5 m and the initial coin velocities ranging from 0.01 to 0.1 m/s and initial rotation ranging from 0.5 to 5 turns/s, ${P}_{edge}=$ 1.17 × 10

^{−3}or approximately 1 throw every 856.

#### 3.3. Inelastic and Perfectly Smooth Bodies

^{−3}or approximately 1 throw every 200 will deliver the coin on its edge.

#### 3.4. Elastic and Perfectly Smooth Bodies

^{−3}or approximately 1 throw every 303 will end with the coin on its edge, i.e., less than in the first two cases.

^{−3}or approximately 1 throw every 300 will deliver the coin on its edge.

#### 3.5. Elastic and Partially Rough Bodies

- -
- the slip is always in the same direction;
- -
- the frictional impulse has a magnitude μN, where μ is the coefficient of friction and a direction opposite to the relative motion of the point of contact on the landing surface.

- -
- both the numerator and denominator are positive if

- -
- both the numerator and denominator are negative if

- -
- if $\mu <\frac{h}{d}$, then condition (72) includes condition (70) and ${\omega}_{0}$ must be positive, i.e., an initial coin counterclockwise rotation;
- -
- if $\mu =\frac{h}{d}$, then condition (73) is the limiting case of conditions (70) and (71), and ${\omega}_{0}$ is positive, i.e., an initial coin counterclockwise rotation;
- -
- if $\frac{h}{d}<\mu <\frac{2hd}{{d}^{2}-{h}^{2}}$, then condition (74) includes condition (71) and ${\omega}_{0}$ is positive, i.e., an initial coin counterclockwise rotation;
- -
- if $\mu >\frac{2hd}{{d}^{2}-{h}^{2}}$, then condition (74) includes Equation (71) and ${\omega}_{0}$ can be either positive or negative, i.e., an initial coin counterclockwise rotation or clockwise rotation.

^{−3}m, which leaves a relatively limited range of values for the four initial parameters. This shows that a coin ending on its edge is a very rare event in reality.

^{−3}or 1 throw approximately every 100 that would end with the coin on its edge.

^{−4}or 1 throw every 5017 ending with the coin on its edge.

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- MacMillan, W.D. Dynamics of Rigid Bodies; Dover Publ.: New York, NY, USA, 1960; pp. 301–303. [Google Scholar]
- Grant, D.F. Chambers Science and Technology Dictionary; Walker, P.M.B., Ed.; Chambers Ltd.: Edinburgh, UK, 1991; p. 455. [Google Scholar]
- Pletser, V. Is the probability of tossing a coin really 50–50%? Part 2: Dynamic Model with rebounds. Preprints, 2022; submitted. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/foundations2030037

**AMA Style**

Pletser V.
Is the Probability of Tossing a Coin Really 50–50%? Part 1: Static Model and Dynamic Models without Rebounds. *Foundations*. 2022; 2(3):547-560.
https://doi.org/10.3390/foundations2030037

**Chicago/Turabian Style**

Pletser, Vladimir.
2022. "Is the Probability of Tossing a Coin Really 50–50%? Part 1: Static Model and Dynamic Models without Rebounds" *Foundations* 2, no. 3: 547-560.
https://doi.org/10.3390/foundations2030037