In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In theoretical studies, the assumption that a coin is fair is often made by referring to an ideal coin.
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If a cheat has altered a coin to prefer one side over another (a biased coin), the coin can still be used for fair results by changing the game slightly. John von Neumann gave the following procedure:[1]
The reason this process produces a fair result is that the probability of getting heads and then tails must be the same as the probability of getting tails and then heads, as the coin is not changing its bias between flips and the two flips are independent. By excluding the events of two heads and two tails by repeating the procedure, the coin flipper is left with the only two remaining outcomes having equivalent probability. This procedure only works if the tosses are paired properly; if part of a pair is reused in another pair, the fairness may be ruined.
Some coins have been alleged to be unfair when spun on a table, but the results have not been substantiated or are not significant.[2] There are statistical procedures for checking whether a coin is fair.
The probabilistic and statistical properties of coin-tossing games are often used as examplars in both introductory and advanced text books and these are mainly based in assuming that a coin is fair or "ideal". For example, Feller (1968) uses this basis to introduce both the idea of random walks and to develop tests for homogeneity within a sequence of observations by looking at the properties of the runs of identical values within a sequence. The latter leads on to a runs test. A time-series consisting of the result from tossing a fair coin is called a Bernoulli process.