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Coin Flip

January 8, 2024

Dice were first used in prehistoric times. The most famous phrase involving dice, "alea iacta est (the die is cast)", was spoken by Julius Caesar (100 BC- 44 BC) when he brought his troops across the Rubicon river into Italy to become dictator of the Roman Republic.

Julius Caesar (100 BC- 44 BC)

Julius Caesar (100 BC- 44 BC).

Caesar was assassinated on the Ides of March (March 15), 44 BC, by a group of Roman Senators. This group of Senators included his friend, Marcus Junius Brutus (c. 85 BC - 42 BC), leading to the famous quotation in Act 3, Scene 1, of William Shakespeare's play, Julius Caesar, "Et tu, Brute (And you, Brutus)."

I was confused as a child when Popeye's nemesis, Bluto, suddenly became Brutus. This was a consequence of an incorrect idea about the copyright owner of the name, Bluto.

(Portion of a Wikimedia Commons image of a marble bust by Andrea Ferrucci (1465-1526), from the Metropolitan Museum of Art)

There are always some people who attempt to skew the chance of success to their advantage. That's why we see many cases of insider trading and cryptocurrency scams, today, and the use of loaded dice. In theory, any of the six cube faces of a die should land with equal probability when tossed, and this concept is used in the dice game, craps, and many board games. A loaded die is a die that's been modified, usually by the insertion of weights, to enhance the probability of a fall onto a specific face. Dice are so frequently used that there are Unicode symbols (U+2680 to U+2685) for their six faces.

A 120-sided die

Left, a photograph of a 120-sided die. Right, a template to create a 120-sided polyhedron by paper folding. I once created a dodecahedron (12-sided polyhedron) from copper-clad printed circuit board material using a similar template for an art project, but I would never attempt a 120-sided figure. (Left image, a 120-sided die by Wintermute115, and right image by Tilman Piesk, both from Wikimedia Commons.)

While computer pseudorandom number generators are presently the best option for making an arbitrary yes/no decision, it was common in the past to flip a fair coin; that is, a coin that will land either heads or tails with equal probability. Just as for loaded dice, a weighted coin can skew this probability. Mathematician, John Edmund Kerrich (1903-1985), discovered that a coin-sized wooden disk, coated on one side with lead, would land on a surface with the wooden side up 67.9% of the time when flipped.[1] However, such a coin, flipped and caught in mid-flight, acts as a fair coin. The mechanics of bouncing on the surface is what biases a weighted coin.

As can be seen in the figure, below, even an unweighted coin has sufficiently different decorations on its two sides that might have a slight influence the probability of landing heads or tails. Another influence might be the initial upwards facing side before the coin flip. The final coin side probability of 46 different currency denominations was determined with respect to their initial state in a experiment of 350,757 coin flips by a huge international research team.[2-3] In this case, coin flipping was defined as flipping a coin into the air with your thumb, and then catching it in your hand.[2]

US Quarter Coin (New Jersey)

The state quarter coin for New Jersey, issued on May 17, 1999. This coin commemorates George Washington's crossing the Delaware River on the night of December 25-26, 1776. (Heads side and tails side images of the New Jersey State quarter coin, both from Wikimedia Commons. Click for larger image.)

Team members were from the University of Amsterdam (Amsterdam, Netherlands), the University of Kassel (Kassel, Germany), the Katholieke Universiteit Leuven (Leuven, Belgium), Georg August University of Göttingen (Göttingen, Germany), Hasselt University (Hasselt, Belgium), the Gutenberg School Wiesbaden (Wiesbaden, Germany), Justus Liebig University Giessen (Giessen, Germany), the University of Zürich (Zürich, Switzerland), the Utrecht University (Utrecht, Netherlands), CNRS, Lyon Neuroscience Research Center (Lyon, France), Vrije Universiteit Amsterdam (Amsterdam, Netherlands), Radboud University (Nijmegen, Netherlands), and ELTE Eotvos Lorand University (Budapest, Hungary).[2]

The experiment was done to test the theory of human coin tossing published in 2007 by Persi Diaconis, Susan Holmes, and Richard Montgomery, known as the D-H-M Model.[2-3] The model predicts that an ordinary coin tends to land on the same side it started when flipped.[2-3] The D-H-M model predicts just a small probability difference, 51/49, as compared with the presumed 50/50 outcome.[2] The presumed physical process is that a flipped coin will spend more time in flight with its initial side facing up, and this makes landing on that side more likely.[3]

Probability of coins landing on the same side from which they've started.

Probability of coins landing on the same side from which they've started. The experimental data confirm the prediction from the Diaconis, Holmes, and Montgomery (D-H-M) Model of coin flipping.[4]

(Potion of fig. 1 of ref. 2.[2] Click for larger image.)

The experiment, which involved 48 people flipping coins minted in 46 countries and making a total of 350,757 coin flips, showed that vigorously flipped coins have a slightly larger probability (0.508 at the 95% confidence level) of landing on the same side as flipped.[2,4] Most interestingly, some people uniformly achieve a 50/50 split when flipping coins, contrary to the D-H-M model.[2-3] So, how significant is this finding? If you flip a coin a 1,000 times, betting $1 each time, your knowledge of this result might earn you $19.[3] A reasonable cycle time of four flips per minute will pay you $4.56/hour. The minimum wage in Tikalon's state of New Jersey is $14.13/hour.


  1. J. E. Kerrich, "An experimental introduction to the theory of probability," (1946, Einar Munksgaard: Copenhagen, Denmark). Unfortunately, not freely accessible.
  2. František Bartoš, Alexandra Sarafoglou, Henrik R. Godmann, Amir Sahrani, David Klein Leunk, Pierre Y. Gui, David Voss, Kaleem Ullah, Malte J. Zoubek, Franziska Nippold, Frederik Aust, Felipe F. Vieira, Chris-Gabriel Islam, Anton J. Zoubek, Sara Shabani, Jonas Petter, Ingeborg B. Roos, Adam Finnemann, Aaron B. Lob, Madlen F. Hoffstadt, Jason Nak, Jill de Ron, Koen Derks, Karoline Huth, Sjoerd Terpstra, Thomas Bastelica, Magda Matetovici, Vincent L. Ott, Andreea S. Zetea, Katharina Karnbach, Michelle C. Donzallaz, Arne John, Roy M. Moore, Franziska Assion, Riet van Bork, Theresa E. Leidinger, Xiaochang Zhao, Adrian Karami Motaghi, Ting Pan, Hannah Armstrong, Tianqi Peng, Mara Bialas, Joyce Y.-C. Pang, Bohan Fu, Shujun Yang, Xiaoyi Lin, Dana Sleiffer, Miklos Bognar, Balazs Aczel, and Eric-Jan Wagenmakers, "Fair coins tend to land on the same side they started: Evidence from 350,757 flips," arXiv, October 10, 2023.
  3. Bob Yirka, "Flipped coins found not to be as fair as thought," Phys.org, October 11, 2023
  4. Persi Diaconis, Susan Holmes, and Richard Montgomery, "Dynamical bias in the coin toss," SIAM Review, vol. 49, no. 2 (2007), pp. 211-235, https://doi.org/10.1137/S003614450444643. A PDF file can be found here.
  5. František Bartoš, "How fair is a fair coin flip?," Open Science Framework Website, November, 2022.

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