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hubie [soylentnews.org] writes:

An a short but interesting paper recently posted to arXiv [arxiv.org] finds when most people flip a coin, it tends to land on the same side from which the toss started. Their observations are based upon analysis of 350,757 coin flips.

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 or ships') already in Roman times.

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 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); 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); in 2003, a coin toss decided which of two companies would be awarded a public project in Toronto; and in 2004 and 2013, a coin flip was used to break the tie in local political elections in the Philippines.

[...] The standard model of coin flipping was extended by Diaconis, Holmes, and Montgomery (D-H-M) who proposed that when people flip an 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 the D-H-M 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').

Their analysis agrees with the D-H-M model that suggests a coin will land 51 percent of the time on the same side it started.

An interesting tidbit mentioned was that even notable statisticians such as Karl Pearson [st-andrews.ac.uk] did their own experiments:

Throughout history, several researchers have collected thousands of coin flips. In the 18th century, the famed naturalist Count de Buffon collected 2,048 uninterrupted sequences of 'heads' in what is possibly the first statistical experiment ever conducted. In the 19th century, the statistician Karl Pearson flipped a coin 24,000 times to obtain 12,012 tails. And in the 20th century, the mathematician John Kerrich flipped a coin 10,000 times for a total of 5,067 heads while interned in Nazi-occupied Denmark. These experiments do not allow a test of the D-H-M model, however, 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 (i.e., 10,231/20,000 heads by Larwood and 10,014/20,000 tails by Ku) do not provide compelling evidence for or against the D-H-M hypothesis.

So now that you know how to bet, will this help you make money?

Could future coin tossers use the same-side bias to their advantage? 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. This is more than the casino advantage for 6 deck blackjack against an optimal-strategy player, where the casino would make 5 dollars on a comparable bet, but less than the casino advantage for single-zero roulette, where the casino would make 27 dollars on average. 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.

arXiv paper: arXiv:2310.04153 [arxiv.org] [math.HO]

Original Submission