Question

One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is white.

Answer 1

A white ball can be drawn in two mutually exclusive ways:
(I) By transferring a black ball from first to second bag, then drawing a white ball
(II) By transferring a white ball from first to second bag, then drawing a white ball
Let E₁, E₂ and A be the events as defined below:
E₁ = A black ball is transferred from first to second bag
E₂ = A white ball is transferred from first to second bag
A = A white ball is drawn

$$\therefore P\left(E_1\right)=\frac{5}{9}$$
$$P\left(E_2\right)=\frac{4}{9}$$
$$\text{ Now },$$
$$P\left(A/E_1\right)=\frac{6}{14}$$
$$P\left(A/E_2\right)=\frac{7}{14}$$
$$\text{ Using the law of total probability, we get }$$
$$\text{ Required probability }=P\left(A\right)=P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right)$$
$$=\frac{5}{9}\times \frac{6}{14}+\frac{4}{9}\times \frac{7}{14}$$
$$=\frac{30}{126}+\frac{28}{126}$$
$$=\frac{58}{126}=\frac{29}{63}$$