Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and n white balls. One bag…

Let $E_1$ event denotes the selection for $1^{\mathrm{st}}$ bag

and $E_2$ event denotes the selection for $2^{\mathrm{nd}}$ bag

Given $P\left(E_1\right)=\frac{1}{2},P\left(E_2\right)=\frac{1}{2}$

Let the event $A=$ selected balls are $1$ red & $1$ black

$P\left(\frac{A}{E_1}\right)=\frac{{}^{3}C_1\times {}^{1}C_1}{{}^{6}C_2}=\frac{1}{5}$

$P\left(\frac{A}{E_2}\right)=\frac{{}^{3}C_1\times {}^{2}C_1}{{\left(n+5\right)}_{C_2}}=\frac{12}{\left(n+5\right)\left(n+4\right)}$

$P\left(\frac{E_1}{ A}\right)=\frac{P\left(E_1\right)\times P\left(\frac{A}{E_1}\right)}{P\left(E_1\right)\times P\left(\frac{A}{E_1}\right)+P\left(E_2\right)\times P\left(\frac{A}{E_2}\right)}$

$\Rightarrow \frac{\frac{1}{2}\times {\displaystyle \frac{1}{5}}}{\frac{1}{2}\times \frac{1}{5}+\frac{6}{\left(n+5\right)\left(n+4\right)}}=\frac{6}{11}$

$\Rightarrow n=4$