Bag $B_1$  contains 4 white and 2 black balls. Bag $B_{2 }$ contains 3 white and 4 black balls. A bag is  drawn at random and a ball is chosen at random from it. What is the probability that the ball drawn is white ?

Conditional probability : If A & B are two events in a sample space S and $P\left(B\right)\ne 0$, the probability of A given the event B has occurred is called conditional probability of A given B and is denoted by$P(A/B)$, we define

$P(A/B)=\frac{P(A\cap B)}{P\left(B\right)}$

The probability of selecting each bag = $1/2$

$\text{ i.e }P\left(B_1\right)=P\left(B_2\right)=\frac{1}{2}$

The probability of selecting a white ball after

selection of $B_1=P\left(W/B_1\right)=\frac{{ }^4{c}_1}{{ }^6{c}_1}=\frac{2}{3}$

The probability of selecting a white ball after

selection of  $B_2=P\left(W/B_2\right)=\frac{{ }^3{C}_1}{{ }^7{C}_1}=\frac{3}{7}$

Probability of selecting a white ball$=P\left(W\right)=P\left(B_1\right)P\left(W/B_1\right)+P\left(B_2\right)P\left(W/B_2\right)$

$=\frac{1}{2}\frac{4}{6}+\frac{1}{2}\frac{3}{7}=\frac{1}{2}\left(\frac{2}{3}+\frac{3}{7}\right)=\frac{1}{2}\left(\frac{14+9}{21}\right)=\frac{23}{42}$