Lecture3 - 1 CONDITIONAL PROBABILITY Lecture 3...

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1 CONDITIONAL PROBABILITY Lecture 3 ORIE3500/5500 Summer2009 Chen Conditional Probability and Bayes’ Rule 1 Conditional Probability If A and B are two events such that P ( B ) > 0, then the conditional proba- bility of A given B , P ( A | B ) is defined as P ( A | B ) = P ( A B ) P ( B ) . From the definition it is obvious why we make the assumption that P ( B ) > 0(since it is in the denominator). If P ( B ) = 0 then conditional probability of A given B is not defined and for computations you can take it to be any number. But any how note that P ( A B ) = P ( B ) P ( A | B ) (1) always holds. If P ( B ) > 0 then it follows from the definition of conditional probability and if P ( B ) = 0 then both sides of the equation equals 0. When this is generalized to higher number of events the phenomenon is often called the chain rule . Theorem (Chain Rule) . If A 1 ,A 2 ,...,A n are events, then P ( A 1 ∩ ··· ∩ A n ) = P ( A 1 ) P ( A 2 | A 1 ) P ( A 3 | A 1 A 3 ) ··· P ( A n | A 1 ∩ ··· A n - 1 ) . It is not difficult to see how we arrive at this result. It follows by repeated application of (1). P ( A 1 ∩ ··· ∩ A n ) = P ( A 1 ∩ ··· A n - 1 ) P ( A n | A 1 ∩ ··· A n - 1 ) = P ( A 1 ∩ ··· A n - 2 ) P ( A n - 1 | A 1 ∩ ··· A n - 2 ) · P ( A n | A 1 ∩ ··· A n - 1 ) . . . = P ( A 1 ) P ( A 2 | A 1 ) P ( A 3 | A 1 A 3 ) ··· P ( A n | A 1 ∩ ··· A n - 1 ) . 1
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1 CONDITIONAL PROBABILITY Conditional Probabilities satisfy the probability law Conditional probabilities are very much similar to normal probabilities. It is like you restrict your attention to the event B and scale up the probabilities of the subsets of B so that it becomes a probability itself when you consider B to be your sample space. 1. (Nonnegativity) For any
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Lecture3 - 1 CONDITIONAL PROBABILITY Lecture 3...

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