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cs-401r:axioms-of-probability-theory [2014/09/05 16:55] cs401rpml [Third Axiom] |
cs-401r:axioms-of-probability-theory [2014/09/05 17:02] cs401rpml [Third Axiom] |
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== First Axiom == | == First Axiom == | ||

The probability of any event $E$ is between 0 and 1: | The probability of any event $E$ is between 0 and 1: | ||

- | \begin{equation} | + | |

- | 0 \leq P\left(E\right) \leq 1 | + | $0 \leq P\left(E\right) \leq 1$ |

- | \end{equation} | + | |

== Second Axiom == | == Second Axiom == | ||

The probability of the entire sample space $\Omega$ (equivalently, the "certain event") is 1. | The probability of the entire sample space $\Omega$ (equivalently, the "certain event") is 1. | ||

- | \begin{equation} | + | |

- | P\left(\Omega\right) = 1 | + | $P\left(\Omega\right) = 1$ |

- | \end{equation} | + | |

== Third Axiom == | == Third Axiom == | ||

If for all $0 \leq i, j$ such that $i \neq j$ the events $A_i$ and $A_j$ are disjoint (i.e., $A_i \cap A_j = \emptyset$), then | If for all $0 \leq i, j$ such that $i \neq j$ the events $A_i$ and $A_j$ are disjoint (i.e., $A_i \cap A_j = \emptyset$), then | ||

- | \begin{equation} | + | |

- | P\left(\bigcup_{j=1}^{\infty}{A_j}\right) = \sum_{j=1}^{\infty}{P\left(A_j\right)} | + | $P\left(\bigcup_{j=1}^{\infty}{A_j}\right) = \sum_{j=1}^{\infty}{P\left(A_j\right)}$ |

- | \end{equation} | + | |

In the simple case of two variables, if $A \cap B = \emptyset$ then: | In the simple case of two variables, if $A \cap B = \emptyset$ then: | ||

- | \begin{equation} | + | |

- | P\left(A \cup B\right) = P\left(A\right) + P\left(B\right) | + | $P\left(A \cup B\right) = P\left(A\right) + P\left(B\right)$ |

- | \end{equation} | + | |