Question

From each of the three boxes containing $3$ white and $1$ black, $2$ white and $2$ black, $1$ white and $3$ black balls, one ball is drawn at random. If $\frac{p}{q}$ represents the probability that $2$ white and $1$ black balls are drawn then evaluate $3p-q$ . [Given that G.C.D. $\left(\right.p,q\left.\right)=1$ ].

Answer 1

$I:3W,1B$
$II:2W,2B$
$III:1W,3B$
$W_{i}=$ the event of drawing a white ball from
$i^{t h}$ box, $i=1,2,3$
$B_{i}=$ the event of drawing a black ball from
$i^{t h}$ box, $i=1,2,3$
$P\left(2 W , 1 B\right)=P\left(W_{1} \cap W_{2} \cap B_{3}\right)+P\left(W_{1} \cap B_{2} \cap W_{3}\right)+P\left(B_{1} \cap W_{2} \cap W_{3}\right)$
$=P\left(W_{1}\right)P\left(W_{2}\right)P\left(B_{3}\right)+P\left(W_{1}\right)P\left(B_{2}\right)P\left(W_{3}\right)+P\left(B_{1}\right)P\left(W_{2}\right)P\left(W_{3}\right)$
$=\frac{3}{4}\times \frac{2}{4}\times \frac{3}{4}+\frac{3}{4}\times \frac{2}{4}\times \frac{1}{4}+\frac{1}{4}\times \frac{2}{4}\times \frac{1}{4}$
$=\frac{26}{64}=\frac{13}{32}$
$\Rightarrow p=13,q=32$
$\Rightarrow 3p-q=39-32=7$