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Lec 1C

# Lec 1C - Conditional Probability Definition P(AIB = the...

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Page 1 Conditional Probability Definition P (A I B) = the conditional probability of A given B = the probability that A occurs given B has occurred = the probability that A occurs in the sample space of event B A B P(B I A) ? ( ± ? ) ? ( ± ) ≠ P( A I B) P(A I B) ? ( ± ? ) ? ( ? ) or ? ( ± ? ) = P(A I B) ? ( ? )

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Page 2 Bayes’ Formula Formula: Proof: – First express the numerator of the conditional probability of A given B in terms of the conditional probability of B given A using the commutative property of intersections: – Then express the probability of B in the denominator in terms of the conditional probabilities of A and A’. Note that the union of A and A’ is the complete sample space A U A’=S P(A I B) = ? ?±²³? ( ² ) ? ?±²³? ²³ + ? ?±² ³? ( ² ) ² ? = ? ² P(A I B) ? ( ² ? ) ? ( ? ) = ? ( ? ² ) ? ( ? ) = ? ?±²³? ( ² ) ? ( ? )
Page 3 Bayes’ Formula (Con’t) Proof (Con’t): – Substituting the above into the result of the first step one gets the Bayes’ formula P(A I B) = ? ?±² ³ ? ( ² ) ? ?±² ³ ? ² ³ + ? ?±² ³ ? ( ² ) A

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Lec 1C - Conditional Probability Definition P(AIB = the...

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