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10 November 2023

How fair is a coin toss?

The answer, according to a preprint paper by a team of University of Amsterdam researchers, is not the 50-50 odds that statistical theory would predict for a coin flip.

They found the answer after having 48 humans flip real coins from 46 different currencies and denominations a total of 350,757 times. They found the side of the coin that was originally facing up before being flipped would show up again as the result of the coin flip approximately 50.8% percent of the time, or in statistical terms, they found Pr(same side) = 0.508. They also found a 95% confidence interval for their of [0.506, 0.509].

Interestingly, they didn't find a bias in favor of either "Heads" or "Tails" when flipping coins. Those odds did come out even, as the probability of coming up heads was very close to 50%, or Pr(heads) = 0.500 with a 95% confidence interval of [0.498, 0.502].

Although the same-side bias odds are close to even, it's also enough of a bias that coin flip odds are skewed as much as some casino games are in favoring the house. Here's how corresponding author František Bartoš described it:

The natural question to ask next is why are coin flips biased this way. The researchers discuss what they believe is behind the phenomenon:

This variability is consistent with D-H-M model, in which the same-side bias originates from off-axis rotations (i.e., precession or wobbliness), which can reasonably be assumed to vary between people. Future work may attempt to verify whether ‘wobbly tossers’ show a more pronounced same-side bias than ‘stable tossers’. The effort required to test this more detailed hypothesis appears to be excessive, as it would involve detailed analyses of high-speed camera recordings for individual flips.

The D-H-M model refers to a 2007 study by Persi Diaconis, Susan Holmes, and Richard Montgomery that identified the role of the laws of mechanics in determining the outcome of a coin toss based on its initial condition. They concluded in their study "coin tossing is 'physics' not 'random'".

The University of Amsterdam researcher identify a trick for how to make a coin toss fair for when its outcome makes a real world difference. The trick is to conceal which side of the coin is facing up from human observers before the coin is flipped.

The findings are surprising enough that we've built a tool that can replicate the results of human coin flipping based on the study's results. The following tool will let you set the probability of an outcome of "Heads" assuming that is always the side facing up before the coin is flipped, where we've set the default value to 50.8%. You're welcome to run through as many of these 'physics-free' simulations as you like to compare the results with physical coin flips. [If you're reading this post on a site that republishes our RSS news feed, you may need to click through to our site to access a working version of the tool.]

Note: Just as in real life, you may get several Heads or Tails in a row as you click the "Flip Coin" button before you see the result change.

Until other research says otherwise, this may be the most true-to-life coin flip simulator you can ever expect to find!

### References

František Bartoš, et al. Fair coins tend to land on the same side they started: Evidence from 350,757 flips. [PDF document]. DOI: 10.48550/arXiv.2310.04153. 10 October 2023.

Persi Diaconis, Susan Holmes, and Richard Montgomery. Dynamical bias in the coin toss. SIAM Review. 2007; 49(2): 211–235. DOI: 10.1137/S003614450444643. [Ungated PDF document].

Bing Chat. Basic code for this tool was generated using the prompt: "Please write the JavaScript and HTML code for a calculator that can indicate the result of a coin flip. The calculator should let the user indicate the percentage of coin flips that will have "Heads" as a result. The calculator should provide an image of either the Heads or Tails side of the coin as part of its output." We made minor modifications to the generated code.

Image credit: U.S. Mint Coin Classroom. Half Dollar. [Online article]. United States Government works. Accessed 5 November 2023.

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