Bag A contains 3 white, 7 red balls and bag B contains 3 white, 2...

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 $E$: white ball is drawn

Using Baye's theorem,
$⇒ P\left(\dfrac{{E}_{1}}{E}\right) = \dfrac{P ( {E}_{1} ) \cdot P\left(\dfrac{E}{{E}_{1}}\right)}{P\left({E}_{1}\right) \cdot P\left(\dfrac{E}{{E}_{1}}\right) + P\left({E}_{2}\right) \cdot P\left(\dfrac{E}{{E}_{2}}\right)}$

$⇒ P\left(\dfrac{{E}_{1}}{E}\right) = \dfrac{\dfrac{1}{2} \times \dfrac{3}{10}}{\dfrac{1}{2} \times \dfrac{3}{10} + \dfrac{1}{2} \times \dfrac{3}{5}}$

$⇒ P\left(\dfrac{{E}_{1}}{E}\right) = \dfrac{\dfrac{3}{20}}{\dfrac{3}{20} + \dfrac{6}{20}}$

$⇒ P\left(\dfrac{{E}_{1}}{E}\right) = \dfrac{3}{3 + 6}$

$⇒ P\left(\dfrac{{E}_{1}}{E}\right) = \dfrac{1}{3}$