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In its most elementary form, the conditional probability P( A| B) of an event A given an event B is defined by \(\displaystyle P(A| B) = \frac {B(A \cap B)}{P(B)}, \) provided that P( B) ≠ 0. An immediate consequence of the definition is Bayes’ theorem: if A 1, A 2, …, A n are mutually disjoint events whose union has probability one, then \(P(A_1|B) = \frac {P(B|A_1) P(A_1)}{\sum _{i=1}^n P(B|A_i) P(A_i)}\) .

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Conditional Expectation and Probability

  • 工具书词条

  • 发表日期：

    2025年

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  • Takis Konstantopoulos

摘要

In its most elementary form, the conditional probability P( A| B) of an event A given an event B is defined by \(\displaystyle P(A| B) = \frac {B(A \cap B)}{P(B)}, \) provided that P( B) ≠ 0. An immediate consequence of the definition is Bayes’ theorem: if A 1, A 2, …, A n are mutually disjoint events whose union has probability one, then \(P(A_1|B) = \frac {P(B|A_1) P(A_1)}{\sum _{i=1}^n P(B|A_i) P(A_i)}\) .