Question

A bag contains $3$ white and $2$ black balls and another bag contains $2$ white and $4$ black balls. One bag is chosen at random. From the selected bag, one ball is drawn. Find the probability that the ball drawn is white.

Answer 1

Consider the following events:
$E_1=$Selecting first bag.

$E_2=$Selecting second bag ,

$A=$ ball draw is white.

Then $P\left(E_1\right)=P\left(E_2\right)=\frac{1}{2},P\left(A/E_1\right)=\frac{3}{5},P\left(A/E_2\right)=\frac{2}{6}$

Required probability $=P\left(A\right)=P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right)$

$=\frac{1}{2}\times \frac{3}{5}+\frac{1}{2}\times \frac{2}{6}$

$=\frac{7}{15}$