A bag contains 4 white and 5 black balls. Another bag contains 9 white and 7 black balls. A ball is transferred from the first bag to the second and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.

Here, ${W_1} = \{ 4$ white balls $\}$ and ${B_1} = \{ 5$ black balls $\}$

and ${W_2} = \{ 9$ white balls $\}$ and ${B_2} = \{ 7$ black balls $\}$

Let ${E_1}$ is the event that ball transferred from the first bag is white and ${E_2}$ is the event that the ball transferred from the first bag is black.

Also, $E$ is the event that the ball drawn from the second bag is white.

$\therefore$ $P\left( {E/{E_1}} \right) = \frac{{10}}{{17}},P\left( {E/{E_2}} \right) = \frac{9}{{17}}$

and $P\left( {{E_1}} \right) = \frac{4}{9}$

and $P\left( {{E_2}} \right) = \frac{5}{9}$

$\therefore$ $P(E) = P\left( {{E_1}} \right) \cdot P\left( {E/{E_1}} \right) + P\left( {{E_2}} \right) \cdot P\left( {E/{E_2}} \right)$

$= \frac{4}{9} \cdot \frac{{10}}{{17}} + \frac{5}{9} \cdot \frac{9}{{17}}$

$= \frac{{40 + 45}}{{153}} = \frac{{85}}{{153}} = \frac{5}{9}$