Question
Let $$A$$ and $$B$$ be two independent events such that $$P\left( A \right) = \frac{1}{5},\,P\left( {A \cup B} \right) = \frac{7}{{10}}.$$ Then $$P\left( {\overline B } \right)$$ is equal to :
A.
$$\frac{3}{8}$$
B.
$$\frac{2}{7}$$
C.
$$\frac{7}{9}$$
D.
none of these
Answer :
$$\frac{3}{8}$$
Solution :
$$\eqalign{
& {\text{Let }}P\left( {\overline B } \right) = x.{\text{ Then }}P\left( B \right) = 1 - x \cr
& \therefore \,P\left( {A \cap B} \right) = P\left( A \right)P\left( B \right) = \frac{1}{5}\left( {1 - x} \right) \cr
& {\text{But }}P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right) \cr
& \Rightarrow \frac{7}{{10}} = \frac{1}{5} + \left( {1 - x} \right) - \frac{1}{5}\left( {1 - x} \right){\text{ or }}\frac{1}{2} = \frac{4}{5}\left( {1 - x} \right) \cr
& \therefore \,1 - x = \frac{5}{8}\,\,\,\,\,\,\,\,\,\,\,\therefore \,x = \frac{3}{8} \cr} $$