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Question
The bag A contains 8 white and 7 black balls while the bag B contains 5 white and 4 black balls. One ball is randomly picked up from the bag A and mixed up with the balls in bag B. Then a ball is randomly drawn out from it. Find the probability that ball drawn is white.
Solution
A white ball can be drawn in two mutually exclusive ways:
(I) By transferring a black ball from bag A to bag B, then drawing a white ball
(II) By transferring a white ball from bag A to bag B, then drawing a white ball
Let E1, E2 and A be events as defined below:
E1 = A black ball is transferred from bag A to bag B
E2 = A white ball is transferred from bag A to bag B
A = A white ball is drawn
\[\therefore P\left( E_1 \right) = \frac{7}{15}\]
\[ P\left( E_2 \right) = \frac{8}{15}\]
\[\text{ Now } , \]
\[P\left( A/ E_1 \right) = \frac{5}{10} = \frac{1}{2}\]
\[P\left( A/ E_2 \right) = \frac{6}{10} = \frac{3}{5}\]
\[\text{ Using the law of total probability, we get} \]
\[\text{ Required probability } = P\left( A \right) = P\left( E_1 \right)P\left( A/ E_1 \right) + P\left( E_2 \right)P\left( A/ E_2 \right)\]
\[ = \frac{7}{15} \times \frac{1}{2} + \frac{8}{15} \times \frac{3}{5}\]
\[ = \frac{7}{30} + \frac{8}{25}\]
\[ = \frac{35 + 48}{150} = \frac{83}{150}\]
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