Question

Let $A$ and $B$ be the events such that $P\left(A\right)=\frac{7}{13},P\left(B\right)=\frac{9}{13}$ and $P(A\cap B)=\frac{4}{13}$
Find $P\left(\overline{B}/\overline{A}\right)$

Answer 1

Given $P\left(A\right)=\frac{7}{13},P\left(B\right)=\frac{9}{13},P(A\cap B)=\frac{4}{13}$

As we know that

$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

$⟹P(A\cup B)=\frac{7}{13}+\frac{9}{13}-\frac{4}{13}=\frac{12}{13}$

$P(\overline{A}\cap \overline{B})=1-P(A\cup B)=1-\frac{12}{13}=\frac{1}{13}$

$P\left(\overline{A}\right)=1-P\left(A\right)=1-\frac{7}{13}=\frac{6}{13}$

$P\left(\overline{B}/\overline{A}\right)=\frac{P(\overline{B}\cap \overline{A})}{P\left(\overline{A}\right)}=\frac{\frac{1}{13}}{\frac{6}{13}}=\frac{1}{6}$