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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
1 have equal probability, and W is consists of two of the ﬁve possibilities. Second, by the law of
P (B ) = P (B |W )P (W ) + P (B |W c )P (W c )
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 )
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
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.
- Spring '08
- The Land