Question
For two events $$A$$ and $$B$$ it is given that $$P\left( A \right) = P\left( {\frac{A}{B}} \right) = \frac{1}{4}{\text{ and }}P\left( {\frac{B}{A}} \right) = \frac{1}{2}.$$
Then :
A.
$$A$$ and $$B$$ are mutually exclusive events
B.
$$A$$ and $$B$$ are dependent events
C.
$$P\overline {\left( {\frac{A}{B}} \right)} = \frac{3}{4}$$
D.
none of these
Answer :
$$P\overline {\left( {\frac{A}{B}} \right)} = \frac{3}{4}$$
Solution :
We have, $$P\left( A \right) = P\left( {\frac{A}{B}} \right) = \frac{1}{4}$$
This shows that $$A$$ and $$B$$ are independent events.
$$\eqalign{
& {\text{So, }}P\left( B \right) = P\left( {\frac{B}{A}} \right) = \frac{1}{2} \cr
& {\text{Now, }}P\left( {\frac{A}{B}} \right) = \frac{1}{4} \cr
& \Rightarrow P\overline {\left( {\frac{A}{B}} \right)} = 1 - \frac{1}{4} = \frac{3}{4} \cr} $$