# total probability problem

One bag contains 5 white and 4 black balls. Another bag contains 7 white and 9 black balls. A ball is transferred from the first bag to the second and then a ball is drawn from second. What will be the probability that the ball is white.

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## 1 Answer

It would be the following: $$P(W) = P(W \cap W_1)+P(W \cap B_1)$$

$$= P(W|W_1)P(W_1)+P(W|B_1)P(B_1)$$ $$= \frac{5}{9} \frac{8}{17}+ \frac{4}{9} \frac{7}{17}$$

where $W_1$ denotes the event that the first ball chosen is white, $B_1$ denotes the event that the first ball chosen is black, and $W$ denotes the event the the ball from the second bag is white.

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2
Did you mean $P(W)=P(W|W_1)P(W_1)+P(W|B1)P(B_1)$ ? – Henry Apr 26 '11 at 22:29
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