Coin Tossing

An idealized coin consists of a circular disk of zero thickness which, when thrown in the air and allowed to fall, will rest with either side face up ("heads" H or "tails" T) with equal probability. A coin is therefore a two-sided die. Despite slight differences between the sides and nonzero thickness of actual coins, the distribution of their tosses makes a good approximation to a p=1/2 Bernoulli distribution.

Amazingly, spinning a penny instead of tossing it results in heads only about 30% of the time (Paulos 1995).

Diaconis et al. (2007) proposed that flipping a coin introduces a small degree of wobble which causes the coin to spend more time in the air (before landing) with the initial top side facing up. As a result, a "same-side bias" is introduced so that a coin is slightly more likely to land with the side initially facing upward before the toss on top after landing. Diaconis et al. (2007) estimated a "same side" probability for a fair coin toss of 51% (as opposed to exactly 50%) based on a "modest" number of empirical observations. This prediction has been confirmed experimentally by Bartoš et al. (2013), who collected a total of 350,757 coin flips and found that the probability of the initial side facing upwards upon landing was 50.8% with a 95% confidence interval of 50.6%-50.9%. The data also showed no trace of a heads-tails bias, with 175,420 heads obtained from 350,757 tosses, giving a heads probability of 50.0% with a 95% confidence interval of 49.8%-50.2% (Bartoš et al. 2013).

There are some rather counterintuitive properties of coin tossing. For example, for a fair coin toss, it is twice as likely that the triple TTH will be encountered before THT than after it, and three times as likely that THH will precede HHT. Furthermore, it is six times as likely that HTT will be the first of HTT, TTH, and TTT to occur than either of the others (Honsberger 1979). There are also strings S of Hs and Ts that have the property that the expected wait W(S_1) to see string S_1 is less than the expected wait W(S_2) to see S_2, but the probability of seeing S_1 before seeing S_2 is less than 1/2 (Gardner 1988, Berlekamp et al. 2001). Examples include

1. THTH and HTHH, for which W(THTH)=20 and W(HTHH)=18, but for which the probability that THTH occurs before HTHH is 9/14 (Gardner 1988, p. 64),

2. W(TTHH)=W(THHH)=16, W(HHH)=14, but for which the probability that TTHH occurs before HHH is 7/12, and for which the probability that THHH occurs before HHH is 7/8 (Penney 1969; Gardner 1988, p. 66).


The probability of a coin of finite thickness landing on an edge has been computed by Hernández-Navarro and Piñero (2022) as




is the critical angle of the cylinder of radius r and thickness h comprising the coin. Remarkably, this expression is independent of the coefficient of restitution. The chances of US coins landing on an edge computed using this formula are summarized in the following table.

coindiameter (mm)thickness (mm)P_c

The study of runs of two or more identical tosses is well-developed, but a detailed treatment is surprisingly complicated given the simple nature of the underlying process. For example, the probability that no two consecutive tails will occur in n tosses is given by F_(n+2)/2^n, where F_k is a Fibonacci number. Similarly, the probability that no k consecutive tails will occur in n tosses is given by F_(n+2)^((k))/2^n, where F_l^((k)) is a Fibonacci k-step number.

Toss a fair coin over and over, record the sequence of heads and tails, and consider the number of tosses needed such that all possible sequences of heads and tails of length n occur as subsequences of the tosses. The minimum number of tosses is 2^n+n-1 (Havil 2003, p. 116), giving the first few terms as 2, 5, 10, 19, 36, 69, 134, ... (OEIS A052944). The minimal sequences for n=1 are HT and TH, and for n=2 are HHTTH, HTTHH, THHTT, and TTHHT. The numbers of distinct minimal toss sequences for n=1, 2, ... are 2, 4, 16, 256, ... (OEIS A001146), which appear to simply be 2^(2^(n-1)).

It is conjectured that as n becomes large, the average number of tosses needed to get all subsequences of length n is 2^n(gamma+nln2), where gamma is the Euler-Mascheroni constant (Havil 2003, p. 116).

See also

Bernoulli Distribution, Bernoulli Trial, Cards, Coin, Dice, Gambler's Ruin, Martingale, Run, Saint Petersburg Paradox

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Bartoš, F. et al. "Fair Coins Tend to Land on the Same Side They Started: Evidence From 350,757 Flips." 10 Oct 2023., E. R.; Conway, J. H; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 1: Adding Games, 2nd ed. Wellesley, MA: A K Peters, p. 777, 2001.Diaconis, P.; Holmes, S.; and Montgomery, R. "Dynamical Bias in the Coin Toss." SIAM Review 49, 211-235, 2007.Ford, J. "How Random is a Coin Toss?" Physics Today 36, 40-47, 1983.Gardner, M. "Nontransitive Paradoxes." Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 64-66, 1988.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Hernández-Navarro, L. Piñero , J. "Exact Face-Landing Probabilities for Bouncing Objects: Edge Probability in the Coin Toss and the Three-Sided Die Problem." Phys. Rev. E 105, L022201-1-6, 2022.Honsberger, R. "Some Surprises in Probability." Ch. 5 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 100-103, 1979.Keller, J. B. "The Probability of Heads." Amer. Math. Monthly 93, 191-197, 1986.Paulos, J. A. A Mathematician Reads the Newspaper. New York: BasicBooks, p. 75, 1995.Penney, W. "Problem 95. Penney-Ante." J. Recr. Math. 2, 241, 1969.Peterson, I. Islands of Truth: A Mathematical Mystery Cruise. New York: W. H. Freeman, pp. 238-239, 1990.Sloane, N. J. A. Sequences A000225/M2655, A001146/M1297, A050227, and A052944 in "The On-Line Encyclopedia of Integer Sequences."Spencer, J. "Combinatorics by Coin Flipping." Coll. Math. J., 17, 407-412, 1986.United States Mint. "Coin Specifications.", E. T. and Robinson, G. "The Frequency Distribution of Tosses of a Coin." §90 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 176-177, 1967.

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Coin Tossing

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Weisstein, Eric W. "Coin Tossing." From MathWorld--A Wolfram Web Resource.

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