It is widely believed that a coin toss is a quick and fair way to resolve random disputes. When a coin is flipped, someone calls out heads or tails, and the probability of the coin being heads or tails is 50/50. But what if the probability of heads or tails is not even? A team of 48 researchers in Amsterdam spent several days flipping coins and found that the results of this random game of chance may not be as random as most people think.


At the beginning of this century, three American mathematicians, led by Persi Diaconis, created a coin-flip machine to study a hypothesis. They argue that a coin toss is far from random. Instead, they believe that coin tossing is a physics problem, a seemingly random process that can produce 100% predictable results with just a few tweaks.

Of course, human hands aren't like machines, but studies conclusively show that most people tend to flip a coin slightly toward the up side. Those who clearly gravitate to one side are called "swinging coin flippers." According to calculations, under normal circumstances, the probability of a coin landing heads is 51%.

To empirically test this hypothesis, a team of researchers at the University of Amsterdam embarked on the mind-numbing exercise of flipping a coin. Over many days, 48 ​​different people flipped 46 different coins. In the end, the researchers tossed a total of 350,757 coins.

"We found overwhelming evidence for the 'same-face' deviation predicted by Diaconis and colleagues in 2007," explains Frantisek-Bartos, one of the lead researchers. "The coin is more likely to land heads if it starts with heads, and vice versa. How big is the bias? In our sample, the average estimate is 50.8%, CI [50.6%, 50.9%]."

Perhaps most striking is the difference in same-face bias across coin tossers. Of the 48 participants, only 10 had less than a 50% chance of preferring the downward side. The rest all favored the up side, with some of the "most wobbly" throwers throwing on the same side almost 60 percent of the time.

But does this same-side bias really make sense in a real-world coin toss?

Stephen Woodcock from the School of Mathematics and Physical Sciences at the University of Technology, Sydney, thinks this doesn't mean much. Woodcock was not involved in the new study. The effect sizes cited in the study were so tiny that they would hardly have an impact in any actual coin-flipping scenario, he said.

"I don't want to get too esoteric, it's almost a philosophical question about what randomness is," Woodcock explained in an email. "Even if there was a small effect (let's be honest - even in a huge sample of over 350,000 coin tosses, the observed deviation of 50.8% vs. 49.2% is nothing more than (Just 8 more heads per thousand flips than a true 50/50 ratio) Does this mean anything in the real world? In most cases, coin tossers don't intentionally place the coin on their thumb, which is a strong form of randomization in itself, and this study didn't take that into account."

Woodcock also pointed to a 2009 study that first questioned the randomness of coin tossing. The study had 13 subjects throw as many faces as possible. Each subject was able to manipulate the coin toss easily and skillfully, favoring heads over tails in most coin tosses.

These 2009 findings raise questions about whether participants in the latest study were intentionally or unintentionally manipulating the results to favor heads. In the preprint article, Bartos and colleagues called this a "legitimate concern" because all participants were aware of the hypotheses being tested.

The researchers wrote: "... it cannot be ruled out that some participants were able to manipulate the outcome of the coin toss to produce a same-side bias. Given the nature of the coin tossing process, the evidence in the video recordings, and the precise correspondence between the data and the D-H-M model predictions, we consider this unlikely; this will need to be confirmed in future work."

Ultimately, Bartos and colleagues concluded that,In high-stakes decision problems, the coin flipper would be better off concealing the starting position of the coin.In Woodcock's view, however, most real-world coin tossing scenarios don't offer the opportunity to observe or change the coin's starting position, so these tiny findings are relatively meaningless in everyday life.

Woodcock noted: "I am a qualified football referee and I can honestly say that I have never noticed whether I put a coin head up or head down on my thumb before flipping it."