2.10 - The Law of Total Probability S = sample space B1 B2 Bk partition S that is k Chapter 2 Section 10 Law of Total Probability Bayes Theorem S =

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1 1 Chapter 2, Section 10 © John J Currano, 02/01/2010 Law of Total Probability Bayes’ Theorem ± S = sample space ± B 1 , B 2 , , B k partition S ; that is ± A is any event (any subset of S ) Then A B 1 , A B 2 , , A B k partition A ; that is j i B B B S j i k i i = = = if and 1 U ( ) j i B B B A A j i k i i = = = if ) (A ) (A and 1 U This result follows from the generalized distributive law : () ( ) ( ) k k B A B A B B A S A A = = = L L 1 1 The Law of Total Probability ± S = sample space ± B 1 , B 2 , , B k partition S ± A is any event (any subset of S ) Then A B 1 , A B 2 , , A B k partition A , and P ( A ) = P ( A B 1 ) + ⋅⋅⋅ + P ( A B k ) = P ( B 1 ) P ( A | B 1 ) + ⋅⋅⋅ + P ( B k ) P ( A | B k ) This result follows from: 1. countable additivity (since the A B i partition A ), and 2. the multiplicative law of probability: P ( A B i )= P ( B i ) P ( A | B i ). The Law of Total Probability B 1 , B 2 , , B k partition S; A S P ( A ) = P ( A B 1 ) + ⋅⋅⋅ + P ( A B k ) = P ( B 1 ) P ( A | B 1 ) + ⋅⋅⋅ + P ( B k ) P ( A | B k ) This theorem is useful in the following fairly common situation:

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This note was uploaded on 02/23/2011 for the course MATH 444 taught by Professor Any during the Fall '10 term at Roosevelt.

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2.10 - The Law of Total Probability S = sample space B1 B2 Bk partition S that is k Chapter 2 Section 10 Law of Total Probability Bayes Theorem S =

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