This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 nonzero 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...
View
Full
Document
This note was uploaded on 11/29/2011 for the course MATH 3355 taught by Professor Britt during the Spring '08 term at LSU.
 Spring '08
 Britt
 Probability

Click to edit the document details