Question

A bag $A$ contains $10$ white and $3$ black balls. Another bag $B$ contains $3$ white and $5$ black balls. Two balls are transferred from bag $A$ and put in the bag $B$ and a ball is drawn from bag $B$. Find the probability that the ball drawn is white ball.

Answer 1

Let us consider the problem,

$S_i$ be the event that you selected bag i; $W,B$ that you selected a white or black ball.

So, 'randomly selected' we selecting either bag is equally likely.

$\begin{array}{c}P\left(S_A|B\right)=\frac{P(S_A,B)}{P\left(B\right)}\\ =\frac{P\left(B|S_A\right)P\left(S_A\right)}{P(S_A,B)+P(S_B,B)}\\ =\frac{P\left(B|S_A\right)P\left(S_A\right)}{P\left(B|S_A\right)P\left(S_A\right)+P\left(B|S_B\right)P\left(S_B\right)}\\ =\frac{\left(\frac{3}{5}\right)\left(\frac{1}{2}\right)}{\left(\frac{3}{5}\right)\left(\frac{1}{2}\right)+\left(\frac{7}{12}\right)\left(\frac{1}{2}\right)}\\ =\frac{36}{71}\end{array}$

Hence, the probability of white ball is $\frac{36}{71}$