This method assumes that the experimenter can decide to toss the coin any number of times. Estimator of true probability ( Frequentist approach).The probability that this particular coin is a "fair coin" can then be obtained by integrating the PDF of the posterior distribution over the relevant interval that represents all the probabilities that can be counted as "fair" in a practical sense. The theory of Bayesian inference is used to derive the posterior distribution by combining the prior distribution and the likelihood function which represents the information obtained from the experiment. Initially, the true probability of obtaining a particular side when a coin is tossed is unknown, but the uncertainty is represented by the " prior distribution". Posterior probability density function, or PDF ( Bayesian approach).The results can then be analysed statistically to decide whether the coin is "fair" or "probably not fair". This article illustrates two of them.īoth methods prescribe an experiment (or trial) in which the coin is tossed many times and the result of each toss is recorded. There are many statistical methods for analyzing such an experimental procedure. This article describes experimental procedures for determining whether a coin is fair or unfair. In more rigorous terminology, the problem is of determining the parameters of a Bernoulli process, given only a limited sample of Bernoulli trials. Therefore, any fairness test must only establish a certain degree of confidence in a certain degree of fairness (a certain maximum bias). It is of course impossible to rule out arbitrarily small deviations from fairness such as might be expected to affect only one flip in a lifetime of flipping also it is always possible for an unfair (or " biased") coin to happen to turn up exactly 10 heads in 20 flips. So it might be necessary to test experimentally whether the coin is in fact "fair" – that is, whether the probability of the coin's falling on either side when it is tossed is exactly 50%. Either a specially designed chip or more usually a simple currency coin is used, although the latter might be slightly "unfair" due to an asymmetrical weight distribution, which might cause one state to occur more frequently than the other, giving one party an unfair advantage. It is based on the coin flip used widely in sports and other situations where it is required to give two parties the same chance of winning. The practical problem of checking whether a coin is fair might be considered as easily solved by performing a sufficiently large number of trials, but statistics and probability theory can provide guidance on two types of question specifically those of how many trials to undertake and of the accuracy of an estimate of the probability of turning up heads, derived from a given sample of trials.Ī fair coin is an idealized randomizing device with two states (usually named "heads" and "tails") which are equally likely to occur. In statistics, the question of checking whether a coin is fair is one whose importance lies, firstly, in providing a simple problem on which to illustrate basic ideas of statistical inference and, secondly, in providing a simple problem that can be used to compare various competing methods of statistical inference, including decision theory. ( January 2010) ( Learn how and when to remove this template message) Please help to improve this article by introducing more precise citations. This article includes a list of general references, but it lacks sufficient corresponding inline citations.
0 Comments
Leave a Reply. |