# conditional probability

Let $(\mathrm{\Omega},\U0001d505,\mu )$ be a probability space^{}, and let $X,Y\in \U0001d505$ be events.

The *conditional probability ^{}* of $X$ given $Y$ is defined as

$$\mu (X|Y)=\frac{\mu (X\cap Y)}{\mu (Y)}$$ | (1) |

provided $\mu (Y)>0$. (If $\mu (Y)=0$, then $\mu (X|Y)$ is not defined.)

If $\mu (X)>0$ and $\mu (Y)>0$, then

$$\mu (X|Y)\mu (Y)=\mu (X\cap Y)=\mu (Y|X)\mu (X),$$ | (2) |

and so also

$$\mu (X|Y)=\frac{\mu (Y|X)\mu (X)}{\mu (Y)},$$ | (3) |

which is Bayes’ Theorem.

Title | conditional probability |
---|---|

Canonical name | ConditionalProbability |

Date of creation | 2013-03-22 12:21:54 |

Last modified on | 2013-03-22 12:21:54 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 8 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 60A99 |

Related topic | ConditionalEntropy |

Related topic | BayesTheorem |

Related topic | ConditionalExpectation |