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Law of total probability : ウィキペディア英語版
Law of total probability

In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct events - hence the name.
==Statement==
The law of total probability is〔Zwillinger, D., Kokoska, S. (2000) ''CRC Standard Probability and Statistics Tables and Formulae'', CRC Press. ISBN 1-58488-059-7 page 31.〕 the proposition that if \left\〕 "overall probability" is sometimes used in less formal writings.〔cite book|author=Deborah Rumsey|title=Probability for dummies|url=http://books.google.com/books?id=Vj3NZ59ZcnoC&pg=PA58|year=2006|publisher=For Dummies|isbn=978-0-471-75141-0|page=58}}〕
The law of total probability can also be stated for conditional probabilities. Taking the B_n as above, and assuming C is an event independent with any of the B_n:
:\Pr(A \mid C) = \sum_n \Pr(A \mid C \cap B_n) \Pr(B_n \mid C) = \sum_n \Pr(A \mid C \cap B_n) \Pr(B_n)

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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