Bag A contains 3 white and 5 black balls while bag B contains 4 white and 3 black balls. A ball is selected at random from bag A and put in bag B. If a ball is now selected at random from bag B then the probability that this ball is white ball is...

Let $E_1$ be the event that a white ball is transferred from bag A to bag B.

Let $E_2$ be the event that a black ball is transferred from bag A to bag B.

Let $W$ be the event that a white ball is drawn from bag B.

The probability of transferring a white ball from bag A is:

$P(E_1) = \dfrac{3}{8}$

The probability of transferring a black ball from bag A is:

$P(E_2) = \dfrac{5}{8}$

If $E_1$ occurs, bag B will contain 5 white and 3 black balls. The probability of drawing a white ball is:

$P(W|E_1) = \dfrac{5}{8}$

If $E_2$ occurs, bag B will contain 4 white and 4 black balls. The probability of drawing a white ball is:

$P(W|E_2) = \dfrac{4}{8}$

By the theorem of total probability:

$P(W) = P(E_1)P(W|E_1) + P(E_2)P(W|E_2)$

$P(W) = \left(\dfrac{3}{8}\right)\left(\dfrac{5}{8}\right) + \left(\dfrac{5}{8}\right)\left(\dfrac{4}{8}\right)$

$P(W) = \dfrac{15}{64} + \dfrac{20}{64} = \dfrac{35}{64}$

Answer: $\dfrac{35}{64}$