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– Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability, that the ball drawn is white, is $\frac{\mathrm{p}}{\mathrm{q}}, \operatorname{gcd}(\mathrm{p}, \mathrm{q})=1$, then $\mathrm{p}+\mathrm{q}$ is equal to [JEE MAIN 23 Jan 2026 S II]
(1) 22
(2) 23
(3) 24
(4) 21

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Solution

Event $\mathrm{E}_{1}$ : white ball is transferred
Event $\mathrm{E}_{2}$ : Black ball is transferred

\[ P\left(E_{1}\right)=\frac{6}{6+4}=\frac{3}{5}, P\left(E_{2}\right)=\frac{4}{6+4}=\frac{2}{5} \]

Event X : From bag A , white ball is drawn

\[ \begin{aligned} & P(X)=P\left(E_{1}\right) P\left(\frac{X}{E_{1}}\right)+P\left(E_{2}\right) P\left(\frac{X}{E_{2}}\right) \\ & =\frac{3}{5}\left(\frac{10}{18}\right)+\frac{2}{5}\left(\frac{9}{18}\right)=\frac{30+18}{5 \times 18}=\frac{8}{15}=\frac{p}{q} \\ & \therefore p+q=23 \text { Answer(2) } \end{aligned} \]

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