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class03_6

# class03_6 - 1.017/1.010 Class 6 Conditional Probability and...

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1.017/1.010 Class 6 Conditional Probability and Bayes Theorem Conditional Probability If two events A and B are not independent we can gain information about P ( A ) if we know that an event in B has occurred. This is reflected in conditional probability of A given B , written as P ( A|B ) : ) ( ) ( ) | ( B P AB P B A P = The unconditional probability P ( A ) is often called the a priori probability while the conditional probability P ( A | B ) is often called the a posteriori probability. Note that conditioning may take place in either direction: P ( AB ) = P ( A | B ) P ( B ) = P ( B | A ) P ( A ) Conditional probabilities are valid probability measures that satisfy all the fundamental axioms. If A and B are independent: P ( A|B ) = P ( A ) Example: A = {Algae bloom occurs } B = {Daily average water temperature above 25 deg. C) Obtain probabilities from long record of daily algae and temperature observations: Suppose P ( A ) = 0.01 , P ( B ) = 0.15 , P ( A, B ) = 0.005 Then: 033 . 0 15 . 0 005 . 0 ) ( ) ( ) | ( = = = B P AB P B A P Probability of a bloom increases significantly if we know that temperature is above 25 deg. C.

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class03_6 - 1.017/1.010 Class 6 Conditional Probability and...

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