Question

A bag contains $5$ white and $4$ black balls and another bag contains $7$ white and $9$ black balls. A ball is drawn from the first bag and two balls drawn from the seconds bag. What is the probability of drawing one white and two black balls?

Answer 1

Bag I	Bag II
$4B$	$7W$
$5W$	$9B$
Total balls in bag $I=9$	Total balls in bag $II=16$

Required probability $=\left(\frac{5}{9}\right)\left(\frac{{}^9{C}_2}{{}^{16}{C}_2}\right)+\left(\frac{4}{9}\right)\left(\frac{{}^7{C}_1{\times }^9{C}_1}{{}^{16}{C}_2}\right)$
$=\frac{5}{9}\times \frac{9\times 8}{16\times 15}+\frac{4}{9}\times \frac{7\times 9}{8\times 15}$
$=\frac{1}{6}+\frac{7}{30}$
$=\frac{5+7}{30}=\frac{12}{30}=\frac{2}{5}$