10-14-2023

# Coin tossing is not actually a 50/50 proposition

Contrary to popular belief, a coin toss may no longer be a simple game of chance. A recent study suggests that the outcome of a coin toss may not be a random heads or tails. The odds now favor the side of the coin facing up.

## Same-side bias

Coin tossing is considered to be a typical example of a random phenomenon, but this may no longer be accurate.

According to a recent study led by researchers at the University of Amsterdam, coin tosses are not as random as we thought, and there may be a slight bias towards the side that starts facing up.

The side of the coin that is facing up before the toss has a higher chance of facing up when the coin lands. The experts refer to this as the “same-side bias.”

## How to study coin tossing

The researchers aimed to test the counterintuitive prediction from a physics model of human coin tossing that was previously created by Persi Diaconis, Susan Holmes and Richard Montgomery.

The team collected coins in 46 different currencies and flipped them over 350,000 times. “A group of 48 people (i.e., all but three of the co-authors) tossed coins of 46 different currencies and denominations and obtained a total number of 350,757 coin flips,” the researchers wrote in their paper.

The data showed that the side facing up before the toss tends to return to the same position. This happened 50.8% of the time.

## The D-H-M model

The D-H-M Model was developed in 2007 by Persi Diaconis, Susan Holmes, and Richard Montgomery. They analyzed the natural process of flipping a coin caught in the hand and established that vigorously flipped coins tend to come up the way they started.

According to the model, the limiting chance of this type of outcome depends solely on the angle between the normal to the coin and the angular momentum vector. They put the chance of the coin coming up as started to about 0.51 (51 percent) for natural flips.

The study was published in the journal Society for Industrial and Applied Mathematics.

## Current results

The findings of the present study, led by František Bartoš at the University of Amsterdam, support the 2007 D-H-M model.

“Our data lend strong support to this precise prediction: the coins landed on the same side more often than not,” wrote the researchers.

“Furthermore, the data revealed considerable between-people variation in the degree of this same-side bias.”

The team also confirmed the generic prediction that when people flip an ordinary coin with the initial side up randomly determined, it is equally likely to land heads or tails.

Consequently, the odds of landing heads or tails are the same, provided the side up before the toss was chosen randomly.

## Coin tossing implications

In a series of posts on X (formerly Twitter), Bartos stresses the importance of the same-side bias in gambling.

He posted: “If you bet a dollar on the outcome of a coin toss 1,000 times, knowing the starting position of the coin toss would earn you \$19 on average. This is more than the casino advantage for six-deck blackjack against an optimal player (\$5), but less than that for single-zero roulette (\$27).”

While this study has established a slight bias towards the side that starts face up when tossing a coin, it is not significant enough to affect the outcome of a coin toss in practice.

The preprint of this study has been published on arXiv.

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