Question

Two events $A$ and $B$ are such that
$P\left(A\right)=\frac{1}{4},P\left(A|B\right)=\frac{1}{4}$ and $P\left(B|A\right)=\frac{1}{2}$
Consider the following statements:
(I) $P\left(\overline{A}|\overline{B}\right)=\frac{3}{4}$
(II) $A$ and $B$ are mutually exclusive
(III) $P\left(A|B\right)+P\left(A|\overline{B}\right)=1$
Then

• A. Only (I) is correct
• B. Only (I) and (II) are correct
• C. Only (I) and (III) are correct
• D. Only (II) and (III) are correct

Answer 1

The correct option is A Only (I) is correct

$P\left(\overline{A}|\overline{B}\right)=1-\frac{1}{4}=\frac{3}{4}$

for A to B mutually Exclusive , $P\left(A|B\right)$x$P\left(B|A\right)=1$ but here $P\left(A|B\right)$x$P\left(B|A\right)=\frac{1}{8}$

$\left(III\right)$ is also incorrect because $\left(I\right)$ is the correct property that is

$P\left(\overline{A}|\overline{B}\right)+P\left(A|B\right)=1$