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

The act of flipping a coin has been an age-old concept. It serves as a very simple way to make a hard decision as it provides an equal chance for either outcome. The ratio has always been 50:50. This assumption is fair because all coins come with two sides and it stands an equal chance to turn up on any one side when somebody flips it. However, a study conducted by American mathematician Persi Diaconis revealed that coin tosses were not a 50-50 probability sometime back.

Frantisek Bartos, a PhD candidate studying psychological methods at the University of Amsterdam, conducted a pre-print study on arXiv that built off the original paper from Persi Diaconis. His results aligned with that of Diaconis. He shared on X (formerly Twitter): "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. How large is the bias? In our sample, the mean estimate is 50.8%, CI."

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.