Question

$A$ and $B$ are events such that $P\left(A\cup B\right)=3/4,P\left(A\cap B\right)=1/4,P\left(\bar{A}\right)=2/3$ if $P\left(\bar{A}\cap B\right)$ is $=\frac{a}{b}$, then $a\times b$ is

Answer 1

Answer: 60

$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$;
$\Rightarrow \frac{3}{4}=1-P\left(\bar{A}\right)+P\left(B\right)-\frac{1}{4}$
$\Rightarrow 1=1-\frac{2}{3}+P\left(B\right)$
$\Rightarrow P\left(B\right)=\frac{2}{3}$ ;
Now, $P\left(\bar{A}\cap B\right)=P\left(B\right)-P\left(A\cap B\right)$
$=\frac{2}{3}-\frac{1}{4}=\frac{5}{12}$
$\Rightarrow \frac{5}{12}=\frac{a}{b}$
$\Rightarrow a=5,b=12$
$\therefore a\times b=5\times 12=60$