chap4slidespt2

# chap4slidespt2 - Conditional Probability Laws Bayes formula...

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Conditional Probability Laws Bayes’ formula From deﬁnition: P ( B | A ) = P ( AB ) P ( A ) , or P ( B | A ) = P ( A | B ) P ( B ) P ( A ) Alternative form: P ( B | A ) = P ( A | B ) P ( B ) P ( A | B ) P ( B ) + P ( A | B c ) P ( B c ) ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 33

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Conditional Probability Laws Bayes’ formula Urn 10 White balls 5 Black balls We draw two balls without replacement. Given that he second ball is white, what is the probability that the ﬁrst one was also white? ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 34
Conditional Probability Laws Bayes’ formula Events: W i = “ selecting white ball at the i th draw B i = “ selecting black ball at the i th draw ” = W c i We seek P ( W 1 | W 2 ) . P ( W 1 | W 2 ) = P ( W 2 | W 1 ) P ( W 1 ) P ( W 2 | W 1 ) P ( W 1 ) + P ( W 2 | B 1 ) P ( B 1 ) = 9 14 · 2 3 9 14 · 2 3 + 10 14 · 1 3 = 9 14 ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 35

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Conditional Probability Laws Bayes’ formula Theorem Suppose that B 1 ,B 2 ,...,B n is a partition of S with P ( B i ) > 0 for i = 1 ,...,n . Let A be any event with P ( A ) > 0 . Then, for any k ∈ { 1 } P ( B k | A ) = P ( A | B k ) P ( B k ) n i =1 P ( A | B i ) P ( B i ) ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 36
Conditional Probability Laws Bayes’ formula Example A car rental agency’s ﬂeet consists of: 40% Escorts, 40% Tauruses and 20% Explorers. These are equipped with either Firestone or Goodyear tires: Firestone Goodyear Escort 35% 65% Taurus 55% 45% Explorer 40% 60% A customer selects a car at random: given that the car is equipped with Firestone tires, what is the probability that it is an Explorer?

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chap4slidespt2 - Conditional Probability Laws Bayes formula...

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