Fair coin

Overview
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.

Fair results from a biased coin
If a cheater has altered a coin to prefer one side over another (a biased coin), surprisingly the coin can still be used for fair results by changing the game slightly. John von Neumann gave the following procedure:


 * 1) Toss the coin twice.
 * 2) If the results match, start over, forgetting both results.
 * 3) If the results differ, use the first result, forgetting the second.

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.