# Bayes theorem

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Bayes' Theorem, also known as Bayes' Rule, is used in statistics and probability theory to relate marginal probabilities and conditional probabilities. In the context of Bayesian probability theory, it is used to update degrees of belief (probabilities) given new information.

## Theorem

Bayes' theorem is stated mathematically as

$P(X|Y,I)=\frac{P(Y|X,I)P(X|I)}{P(Y|I)}$,

where X and Y are independent statements, I is available background information, and

P(X | Y,I) is the posterior probability for X given Y and I,

P(Y | X,I) is the likelihood for Y given X and I,

P(X | I) is the prior probability for X given only I, and

P(Y | I) is sometimes called the evidence or probability for Y given only I.

In a scientific context, X may be a hypothesis, and Y may be experimental data. The theorem can then be used to determine the degree of belief in the hypothesis by using the experimental data.

## Applications and Examples

One popular example of the use of Bayes' theorem is the Monty Hall problem, inspired by the television show Let's Make a Deal.