Question

Two urn contains 4 white, 6 blue and 4 white, 5 blue balls. One of the urn is selected at random and a ball is drawn. If the ball drawn is white, Find the probability that it is drawn from the first urn.

Answer 1

Let $E_1$ be the event of choosing bag I and $E_2$ be the event of choosing bag II

A be the event of drawing white ball from first bag.

$P\left(E_1\right)=P\left(E_2\right)=\frac{1}{2}$

$P\left(A/E_1\right)=\frac{4}{10}$

$P\left(A/E_2\right)=\frac{4}{9}$

By bayes theorem

$P\left(E_1/A\right)=\frac{P\left(E_1\right)P\left(A/E_1\right)}{P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right)}$

$=\frac{\frac{1}{2}\times \frac{4}{10}}{\frac{1}{2}\times \frac{4}{10}+\frac{1}{2}\times \frac{4}{9}}$

$=\frac{\frac{1}{2}\times \frac{4}{10}}{\frac{9\times 4+10\times 4}{2\times 9\times 10}}$

$=\frac{4\times 9}{36+40}=\frac{36}{76}=\frac{9}{19}$

$P\left(E_1/A\right)=\frac{9}{19}$