elemprob-fall2010-page22

# elemprob-fall2010-page22 - .30, P(A | M ) = .25, P(W ) =...

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Unformatted text preview: .30, P(A | M ) = .25, P(W ) = .60 and we want P(W | A). From the deﬁnition P(W | A) = P(W ∩ A) . P(A) As in the previous example, P(W ∩ A) = P(A | W )P(W ) = (.30)(.60) = .18. To ﬁnd P(A), we write P(A) = P(W ∩ A) + P(M ∩ A). Since the class is 40% men, P(M ∩ A) = P(A | M )P(M ) = (.25)(.40) = .10. So P(A) = P(W ∩ A) + P(M ∩ A) = .18 + .10 = .28. Finally, P(W | A) = P(W ∩ A) .18 = . P(A) .28 To get a general formula, we can write P(E | F )P(F ) P(E ∩ F ) = P(E ) P(E ∩ F ) + P(E ∩ F c ) P(E | F )P(F ) = . P(E | F )P(F ) + P(E | F c )P(F c ) P(F | E ) = This formula is known as Bayes’ rule. If E and F are independent, knowing E doesn’t help in predicting F . Thus our estimate of the probability of F given E is the same as the probability of F : P(F | E ) = P(F ). This can be rewritten as P(F ∩ E ) = P(F ), P(E ) 22 ...
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