Fair coin

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.

John Edmund Kerrich performed experiments in coin flipping and found that a coin made from a wooden disk about the size of a crown and coated on one side with lead landed heads (wooden side up) 679 times out of 1000. In this experiment the coin was tossed by balancing it on the forefinger, flipping it using the thumb so that it spun through the air for about a foot before landing on a flat cloth spread over a table. Edwin Thompson Jaynes claimed that when a coin is caught in the hand, instead of being allowed to bounce, the physical bias in the coin is insignificant compared to the method of the toss, where with sufficient practice a coin can be made to land heads 100% of the time.

The probabilistic and statistical properties of coin-tossing games are often used as examples in both introductory and advanced text books and these are mainly based in assuming that a coin is fair or "ideal". For example, Feller 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.

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:

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