Lecture 4 - Conditional Probability &amp; Bayes' Rule

# Lecture 4 - Conditional Probability &amp; Bayes' Rule -...

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1 CONDITIONAL PROBABILITY Lecture 4 ORIE3500/5500 Summer2011 Li 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 ) . (1) From the deﬁnition it is obvious why we make the assumption that P ( B ) > 0 (since it is in the denominator). Example 1: Suppose we toss a fair dice twice. What is the probability that the sum of the 2 dice is 8? Suppose I tell you that the ﬁrst die landed on a 3; what is the probability that the sum of the 2 dice is 8? Ans: Let A be the event that the sum of the 2 dice is 8 and B be the event that the ﬁrst die landed on a 3. Then P ( A ) = 5 / 36 and P ( A | B ) = (1 / 36) / (1 / 6) . Think: how to interpret P ( A | B )? 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 A , 0 P ( A | B ). 2. (Normalization)

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Lecture 4 - Conditional Probability &amp; Bayes' Rule -...

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