1 2pn 2 which for xed p grows linearly with n in

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Unformatted text preview: B=“the ball drawn from Box 2 is black” Find P (W ), P (B ), and P (W |B )? Solution: This problem is difficult to work out in one’s head. But following the definition of conditional probability, it is pretty simple. First, P (W ) = 2 because the five possibilities in Step 5 1 have equal probability, and W is consists of two of the five possibilities. Second, by the law of total probability, P (B ) = P (B |W )P (W ) + P (B |W c )P (W c ) 22 33 13 = ·+·=. 55 55 25 2.10. THE LAW OF TOTAL PROBABILITY, AND BAYES FORMULA 51 Third, P (W |B ) = = = P (W B ) P (B ) P (W )P (B |W ) P (B ) 22 4 5·5 13 = 13 . 25 (We just used Bayes formula, perhaps without even realizing it.) For this example it is interesting to compare P (W |B ) to P (W ). We might reason about the ordering as follows. Transferring a white ball (i.e. event W ) makes removing a black ball in Step 2 (i.e. event B) less likely. So given that B is true, we would expect the conditional probability of W to be smaller than the unconditional 4 2 probability. That is indeed the case ( 13 < 5 ). As we’ve seen, the law of total probabil...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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