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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) 21Solution
Event $\mathrm{E}_{1}$ : white ball is transferred
Event $\mathrm{E}_{2}$ : Black ball is transferred\[ \mathrm{P}\left(\mathrm{E}_{1}\right)=\frac{6}{6+4}=\frac{3}{5}, \mathrm{P}\left(\mathrm{E}_{2}\right)=\frac{4}{6+4}=\frac{2}{5} \]
Event X : From bag A , white ball is drawn
\[ \begin{aligned} & \mathrm{P}(\mathrm{X})=\mathrm{P}\left(\mathrm{E}_{1}\right) \mathrm{P}\left(\frac{\mathrm{X}}{\mathrm{E}_{1}}\right)+\mathrm{P}\left(\mathrm{E}_{2}\right) \mathrm{P}\left(\frac{\mathrm{X}}{\mathrm{E}_{2}}\right) \\ & =\frac{3}{5}\left(\frac{10}{18}\right)+\frac{2}{5}\left(\frac{9}{18}\right)=\frac{30+18}{5 \times 18}=\frac{8}{15}=\frac{\mathrm{p}}{\mathrm{q}} \\ & \therefore \mathrm{p}+\mathrm{q}=23 \text { Answer }(2) \end{aligned} \]
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