Question

A bag contains 5 white and 5 black balls. Second bag contains 8 white and 6 black balls. A ball is transferred from the first bag to the second bag. A ball is drawn randomly from the second bag. Find the probability that the ball drawn is white.

Answer 1

Here,$W_1$ = {5 white balls} and $B_1$= {5 black balls}
And $W_2$ = {8 white balls} and $B_2$= {6 black balls}
Let,
$E_1$ is event that the ball transferred from the first bag is white and
$E_2$is the event that the ball transferred from the bag is black.
Also,$E$ is the event that the ball drawn from the second bag is white.
$P\left(E|E_1\right)=$P(ball drawn from the second bag is white given that ball transferred from the first bag is white)=$\frac{9}{15}$
$P\left(E|E_2\right)=$P(ball drawn from the second bag is white given that ball transferred from the first bag is black)=$\frac{8}{15}$
So, $P\left(E\right)=P\left(E|E_1\right)\times P\left(E_1\right)+P\left(E|E_2\right)\times P\left(E_2\right)$
$P\left(E\right)=\frac{9}{15}\times \frac{1}{2}+\frac{8}{15}\times \frac{1}{2}=\frac{17}{30}$