Question

An urn contains $10$ balls, $4$ red and $6$ blue. A second urn contains $16$ red balls and unknown number of blue balls. A single ball is drawn from each urn. If the probability that both balls are of the same colour is $0.44,$ then the number of blue balls in the second urn is

Answer 1

Let number of blue balls in second urn be $n.$

$P$(ball drawn is of same colour)$=0.44$
$\Rightarrow P\left(RR\right)+P\left(BB\right)=0.44$
$\Rightarrow \frac{4}{10}\cdot \frac{16}{16+n}+\frac{6}{10}\cdot \frac{n}{16+n}=0.44$
$\Rightarrow \frac{2}{5}\cdot \frac{16}{16+n}+\frac{3}{5}\cdot \frac{n}{16+n}=0.44$
$\Rightarrow \frac{32+3n}{5(16+n)}=\frac{44}{100}=\frac{11}{25}\Rightarrow 11\cdot 5(16+n)=25(32+3n)$
$\Rightarrow 55(16+n)=800+75n\Rightarrow 55\cdot 16+55n=800+75n\Rightarrow 880-800=20n\Rightarrow 80=20n$
$\Rightarrow n=4$