Question

For two events $A$ and $B$ if $P\left(A\right)=P\left(\frac{A}{B}\right)=\frac{1}{4}$ and $P\left(\frac{B}{A}\right)=\frac{1}{2}$, then

• A. $A$ is subevent of $B$
• B. $A$ and $B$ are mutually exclusive
• C. $A$ and $B$ independent and $P\left(\frac{A^{\prime }}{B}\right)=\frac{3}{4}$
• D. None of these

Answer 1

Answer: (C)

$A$ and $B$ independent and $P\left(\frac{A^{\prime }}{B}\right)=\frac{3}{4}$

$P\left(A\right)=P\left(\frac{A}{B}\right)=\frac{P(A\cap B)}{P\left(B\right)}\Rightarrow P(A\cap B)=P\left(A\right)P\left(B\right)$
Thus, $A$ and $B$ are independent
Also, $P\left(\frac{A^{\prime }}{B}\right)=\frac{P(A^{\prime }\cap B)}{P\left(B\right)}=\frac{P\left(B\right)-P(A\cap B)}{P\left(B\right)}$
$=1-P\left(\frac{A}{B}\right)=1-\frac{1}{4}=\frac{3}{4}$