Question

If $P\left(A\right)=\frac{1}{4},P\left(B\right)=\frac{1}{5}$ and $P(A\cap B)=\frac{1}{7}$, find $P\left(\overline{A}/\overline{B}\right)$.

Answer 1

Given, $P\left(A\right)=\frac{1}{4},P\left(B\right)=\frac{1}{5}$ and $P(A\cap B)=\frac{1}{7}$.

Therefore,

$P\left(\frac{\overline{A}}{\overline{B}}\right)=\frac{P(\overline{A}\cap \overline{B})}{P\left(\overline{B}\right)}=\frac{1-P(A\cup B)}{1-P\left(B\right)}$

$=\frac{1-\left[P\right(A)+P(B)-P(A\cap B\left)\right]}{1-\frac{1}{5}}$

$=\frac{1-[\frac{1}{4}+\frac{1}{5}-\frac{1}{7}]}{\frac{4}{5}}$

$=\frac{(140-43)}{140}\times \frac{5}{4}=\frac{97}{112}$