Coin flip

A coin flip -- the act of spinning a coin into the air with your thumb and then catching it in your hand -- is often considered the epitome of a chance event. It features as a ubiquitous example in textbooks on probability theory and statistics, and constituted a game of chance ('capita aut navia' -- 'heads of ships') already in Roman times.

Diaconis model

“”A coin flip should land on the same side as it started with a probability of approximately 51%.

According to the standard model of coin flipping, the flip is a deterministic process. The perceived randomness originates from small fluctuations in the initial conditions -- regarding starting position, configuration, upward force, and angular momentum -- combined with narrow boundaries on the outcome space. Therefore, the standard model predicts that when people flip a fair coin, the probability of it landing heads is 50%.

The standard model of coin flipping was extended by Persi Diaconis who proposed that when people flip a ordinary coin, they introduce a small degree of 'precession' or wobble -- a change in the direction of the axis of rotation throughout the coin's trajectory. According to this 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. Based on a modest number of empirical observation (featuring coins with ribbons attached and high-frame-rate video recordings) Diaconis measured the off-axis rotations in typical human flips. Based on these observations, the Diaconis model predicted that a coin flip should land on the same side as it started with a probability of approximately 51%.

Throughout history, several researchers have collected thousands of coin flips. In the 18th century, Count de Buffon collected 2,048 uninterrupted sequences of 'heads'. In the 19th century, Karl Pearson flipped a coin 24,000 times to obtain 12,012 tails. And in the 20th, John Kerrich flipped a coin 10,000 times for a total of 5,067 heads. However, these experiments do not allow a test of the Diaconis model, mostly because it was not recorded whether the coin landed on the same side that it started. A notable exception is a sequence of 40,000 coin flips collected by Janet Larwood and Priscilla Ku in 2009. Larwood always started the flips heads-up and Ku always tails-up. Unfortunately, the results (10,231/20,000 heads by Larwood and 10,014/20,000 tails by Ku) do not provide compelling evidence for or against the Diaconis hypothesis.

Empirical test

“”In order to provide a diagnostic empirical test of the same-side bias hypothesized by Diaconis, we collected a total of 350,757 coin flips. A group of 48 people tossed coins of 46 different currencies x denominations and obtained a total number of 350,757 coin flips. The data confirm the prediction from the Diaconis model, the coins landed how they started more often than 50%. Specifically, the data feature 178,078 same-side landings from 350,757 tosses, Probability(same side) is 0.508, 95% confidence interval between 0.506 - 0.509. The simplicity and perceived fairness of a coin flip, coupled with the widespread availability of coins, may explain why it is often used to make even high-stakes decisions. For example, in 1903 a coin flip was used to determine which of the Wright brothers would attempt the first flight; in 2003, a coin toss decided which off two companies would be awarded a public project in Toronto; in 2004 and 2013, a coin flip was used to break the tie in local political elections in the Philippines; in 1968, a coin flip determined the winner of the European Championship semi-final soccer match between Italy and the Soviet Union (an event which Italy went on to win); and in 1959, a coin flip decided who would get the last plane seat for the tour of rock star Buddy Holly (which crashed and left no survivors) These considerations lead us to suggest that when coin flips are used for high-stakes decision making, the starting position of the coin is best concealed.

—František Bartoš, et al (October 6, 2023) Fair Coins Tend to Land on The Same Side They Started : Evidence From 350,757 Flips arXiv Mathematics History and Overview