Question

One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is white.

Answer 1

A white ball can be drawn in two mutually exclusive ways:
(I) By transferring a black ball from first to second bag, then drawing a white ball
(II) By transferring a white ball from first to second bag, then drawing a white ball
Let E₁, E₂ and A be the events as defined below:
E₁ = A black ball is transferred from first to second bag
E₂ = A white ball is transferred from first to second bag
A = A white ball is drawn

\[\therefore P\left( E_1 \right) = \frac{5}{9} \]
\[ P\left( E_2 \right) = \frac{4}{9}\]
\[\text{ Now } , \]
\[P\left( A/ E_1 \right) = \frac{6}{14}\]
\[P\left( A/ E_2 \right) = \frac{7}{14}\]
\[\text{ Using the law of total probability, we get } \]
\[\text{ Required probability }  = P\left( A \right) = P\left( E_1 \right)P\left( A/ E_1 \right) + P\left( E_2 \right)P\left( A/ E_2 \right)\]
\[ = \frac{5}{9} \times \frac{6}{14} + \frac{4}{9} \times \frac{7}{14}\]
\[ = \frac{30}{126} + \frac{28}{126}\]
\[ = \frac{58}{126} = \frac{29}{63}\]