Bag A contains 3 white, 7 red balls and bag B contains 3 white, 2 red balls. One bag is selected at random…

Given: Given balls in bags are $A\to 3W, 7R$, $B\to 3W, 2R$

Let, $E_1:A$ is selected

, $E_2:B$ is selected

and $\mathrm{E}$: white ball is drawn

Using Baye's theorem,
$\Rightarrow P\left(\frac{E_1}{E}\right)=\frac{\mathrm{P}\left({\mathrm{E}}_1\right)·\mathrm{P}\left({\displaystyle \frac{\mathrm{E}}{{\mathrm{E}}_1}}\right)}{\mathrm{P}\left({\mathrm{E}}_1\right)·\mathrm{P}\left(\frac{\mathrm{E}}{{\mathrm{E}}_1}\right)+\mathrm{P}\left({\mathrm{E}}_2\right)·\mathrm{P}\left(\frac{\mathrm{E}}{{\mathrm{E}}_2}\right)}$

$\Rightarrow P\left(\frac{E_1}{E}\right)=\frac{\frac{1}{2}\times \frac{3}{10}}{\frac{1}{2}\times \frac{3}{10}+\frac{1}{2}\times \frac{3}{5}}$

$\Rightarrow P\left(\frac{E_1}{E}\right)=\frac{\frac{3}{20}}{\frac{3}{20}+\frac{6}{20}}$

$\Rightarrow P\left(\frac{E_1}{E}\right)=\frac{3}{3+6}$

$\Rightarrow P\left(\frac{E_1}{E}\right)=\frac{1}{3}$