Scientists toss 350,757 coins, prove that coin tosses are far from equal odds

Flipping a coin has long been the go-to method for making tough decisions, largely because it’s seen as perfectly fair—each side seemingly has a 50-50 chance. After all, with only two sides, how could there be a bias? However, a groundbreaking study by American mathematician Persi Diaconis suggested otherwise, showing that coin tosses might not actually be as evenly split as we thought.

Building on Diaconis’s research, Frantisek Bartos, a PhD candidate in psychological methods at the University of Amsterdam, conducted a study published on arXiv. Bartos’s research supports Diaconis’s findings, pointing to a subtle bias in coin flips: the starting position matters. In a post on X (formerly Twitter), Bartos explained, “We found overwhelming evidence for a 'same-side' bias predicted by Diaconis and colleagues in 2007: If you start heads-up, the coin is more likely to land heads-up and vice versa.” According to his findings, this “same-side” bias means the coin has a slight edge—around 50.8%—to land on the side it started on.

The probability model, called the "Diaconis Model," changes the way humans have been understanding coin tosses for a long time. According to IFL Science, another team spoke about the model, "According to the Diaconis model, precession causes the coin to spend more time in the air with the initial side facing up. Consequently, the coin has a higher chance of landing on the same side as it started (i.e., 'same-side bias')." The team took a herculean effort and got 48 people to flip 350,757 coins from 46 different countries to come up with their results.

It was found that coins had a 51% chance to land on the same side they were tossed from, the same results Bartos got. Additionally, the team also discovered that the probability of coin tosses was affected by the individual tossing it. Some were shown to favor a certain side, while many others had no such bias. The team came to the conclusion that coin tosses were subtly influenced by the person doing it. While these numbers may not seem huge, they could lead to predictable results in certain scenarios.

Bartos provided an example, saying, "The magnitude of the observed bias can be illustrated using a betting scenario. If you bet a dollar on the outcome of a coin toss (i.e., paying 1 dollar to enter and winning either 0 or 2 dollars depending on the outcome) and repeat the bet 1,000 times, knowing the starting position of the coin toss would earn you 19 dollars on average." They went on to explain how it would play out in a game of blackjack.

They state that this would be more than the advantage that a casino had for a game of blackjack with six decks against a player with optimal strategy. The team then reveals how the casino would make five dollars on a comparable bet, but it would be less than the advantage they had in single-zero roulette, where they would make 27 dollars on average. People who read their study would naturally be curious about how their results would affect a conventional coin toss. They reply to this saying, "When coin flips are used for high-stakes decision-making, the starting position of the coin is best concealed."

You can follow @BartosFra on Twitter for more interesting takes on statistics.

Editor's note: This article was originally published on October 13, 2023. It has since been updated.