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

$P\left(A \cup B\right)= P \left(A\right)+P\left(B\right)-P \left(A \cap B\right)$;
$\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} $
$\Rightarrow a=5, b=12$
$\therefore a \times b = 5\times12=60$