Fair coin
Encyclopedia
In probability theory
and statistics
, a sequence of independent
Bernoulli trial
s 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.
gave the following procedure:
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. There are statistical procedures for checking whether a coin is fair.
s 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
.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, a sequence of independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
Bernoulli trial
Bernoulli trial
In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure"....
s 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.
Fair results from a biased coin
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 NeumannJohn von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
gave the following procedure:
- Toss the coin twice.
- If the results match, start over, forgetting both results.
- 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. There are statistical procedures for checking whether a coin is fair.
Role in statistical teaching and theory
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 walkRandom walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
s and to develop tests for homogeneity
Homogeneity (statistics)
In statistics, homogeneity and its opposite, heterogeneity, arise in describing the properties of a dataset, or several datasets. They relate to the validity of the often convenient assumption that the statistical properties of any one part of an overall dataset are the same as any other part...
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
Bernoulli process
In probability and statistics, a Bernoulli process is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables Xi are identical and independent...
.