and
then
A) A and B are independent
B)
C)
D) All of these
Correct Answer: D
Solution :
\[P\left( \frac{B}{A} \right)=\frac{1}{2}\Rightarrow \frac{P(B\cap A)}{P(A)}=\frac{1}{2}\Rightarrow P(B\cap A)=\frac{1}{8}\] \[P\left( \frac{A}{B} \right)=\frac{1}{4}\,\,\Rightarrow \,\,\frac{P(A\cap B)}{P(B)}=\,\,\frac{1}{4}\Rightarrow \,\,P(B)=\frac{1}{2}\] \[P(A\cap B)=\frac{1}{8}=P(A)\,.\,P(B)\] \[\therefore \] events A and B are independent. Now, \[P\left( \frac{A'}{B} \right)=\frac{P(A'\cap B)}{P(B)}=\frac{P(A')\,P(B)}{P(B)}=\frac{3}{4}\] \[P\left( \frac{B'}{A'} \right)=\,\,\frac{P(B'\cap A')}{P(A')}=\,\,\frac{P(B')\,P(A)}{P(A')}=\frac{1}{2}\]
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