Question

If $A$ and $B$ are two events such that $P\left(A\right)=\frac{1}{4},P\left(B\right)=\frac{1}{2}$ and $P(A\cap B)=\frac{1}{8}$, find $8P\left(\overline{A}and\overline{B}\right)$

Answer 1

It is given that, $P\left(A\right)=\frac{1}{2}$ and $P(A\cap B)=\frac{1}{8}$
$P\left(notonAandnotonB\right)=P(A\prime \cap B\prime )$
$P\left(notonAandnotonB\right)=P\left((A\cup B)\right)\prime [A\prime \cap B\prime =(A\cup B)\prime ]$
$=1-P(A\cup B)$
$=1-[P\left(A\right)+P\left(B\right)-P(A\cap B)]$
$=1-\left[\frac{1}{4}+\frac{1}{2}-\frac{1}{8}\right]$
$=1-\frac{5}{8}$
$=\frac{3}{8}=0.27$