Lecture-6

Lecture-6 - Lecture 6 Bayes’s Formula Review 1 If A is an event such that P A negationslash = 0 and B be another event then the conditional

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Unformatted text preview: Lecture 6. Bayes’s Formula Review 1. If A is an event such that P ( A ) negationslash = 0 and B be another event, then the conditional probability of B given A is: P ( B | A ) = P ( A ∩ B ) P ( A ) . (1) 2. We say A and B are independent if P ( A ∩ B ) = P ( A ) P ( B ) . 3. If A and B are events, we may put the probabilities of the events definable from A and B in a table, as follows: B B c + A x = P ( A ∩ B ) y = P ( A ∩ B c ) x + y = P ( A ) A c z = P ( A c ∩ B ) 1 − x − y − z = P ( A c ∩ B c ) 1 − x − y = P ( A c ) + x + z = P ( B ) 1 − x − z = P ( B c ) 1 Then P ( B | A ) = x x + y . If P ( B ) negationslash = 0, then P ( A | B ) = x x + z . Bayes’s Formula If A and B both have non-zero probability, then equation (1) tells us: P ( A | B ) P ( B ) = P ( A ∩ B ) = P ( B | A ) P ( A ) . From this, we get Bayes’s Formula (simple form): P ( A | B ) = P ( B | A ) P ( A ) P ( B ) . (2) Suppose A 1 , A 2 , ··· , A n are disjoint and B ⊆ A 1 ∪ A 2 ∪ ··· ∪ A n . We have the following Decomposition formula: P ( B ) = P ( A 1 ∩ B ) + P ( A 2 ∩ B ) + ··· + P ( A n ∩ B ) = P ( B | A 1 ) P ( A 1 ) + P ( B | A 2 ) P ( A 2 ) + ··· + P ( B | A n ) P ( A n ) . (3) From (2) and (3) we get Bayes’s Formula: If A 1 , A 2 , ··· , A n are disjoint sets and B ⊆ A 1 ∪ A 2 ∪ ··· ∪ A n , then for any k ∈ { 1 , 2 , . . ., n } : P ( A k | B ) = P ( B | A k ) P ( A k ) ∑ n i =1 P ( B | A i...
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This note was uploaded on 11/29/2011 for the course MATH 3355 taught by Professor Britt during the Spring '08 term at LSU.

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Lecture-6 - Lecture 6 Bayes’s Formula Review 1 If A is an event such that P A negationslash = 0 and B be another event then the conditional

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