# Bayes’ theorem

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states

Let $({A}_{n})$ be a sequence of mutually exclusive events^{} whose union (http://planetmath.org/Union) is the sample space and let $E$ be any event. All of the events have nonzero probability ($P(E)>0$ and $P({A}_{n})>0$ for all $n$). Bayes’ Theorem states

$$P({A}_{j}|E)=\frac{P({A}_{j})P(E|{A}_{j})}{{\sum}_{i}P({A}_{i})P(E|{A}_{i})}$$ |

for any ${A}_{j}\in ({A}_{n})$.

A simpler formulation is:

$$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$ |

For two events, $A$ and $B$ (also with nonzero probability).

## References

- 1 Milton, J.S., Arnold, Jesse C., Introduction to Probability and Statistics: Principles and Applications for Engineering and the Computing Sciences, McGraw Hill, 1995.

Title | Bayes’ theorem |
---|---|

Canonical name | BayesTheorem |

Date of creation | 2013-03-22 12:02:13 |

Last modified on | 2013-03-22 12:02:13 |

Owner | akrowne (2) |

Last modified by | akrowne (2) |

Numerical id | 11 |

Author | akrowne (2) |

Entry type | Theorem |

Classification | msc 60-00 |

Classification | msc 62A01 |

Synonym | Bayes’ Rule |

Related topic | ConditionalProbability |