Question

For two events $A$ and $B,$ if $p\left(A\right)=p\left(A|B\right)=\frac{1}{4}$ and $p\left(B|A\right)=\frac{1}{2}$, then $P\left(\frac{\overline{B}}{\overline{A}}\right)=\frac{m}{n}$ where $m+n=$

Answer 1

We have $p\left(A\right)=p\left(A|B\right)=\frac{p(A\cap B)}{p\left(B\right)}\Rightarrow p(A\cap B)=p\left(A\right)p\left(B\right)$

Therefore $A$ and $B$ are independent

Since $p(A\cap B)=p\left(A\right)p\left(B|A\right)=\left(\frac{1}{4}\right)\left(\frac{1}{2}\right)=\frac{1}{8}\ne 0$

$A$ and $B$ cannot be mutually exclusive

As $A$ and $B$ are independent

$P\left(\frac{\overline{B}}{\overline{A}}\right)=P\left(\overline{B}\right)\Rightarrow P\left(\frac{\overline{B}}{\overline{A}}\right)=1-P\left(B\right)=1-\frac{1}{2}=\frac{1}{2}$