Official solution
\[P\left( {{{{E_1}} \over {{E_2}}}} \right) = {1 \over 2} \Rightarrow {{P({E_1} \cap {E_2})} \over {P({E_2})}} = {1 \over 2}\]
\[P\left( {{{{E_2}} \over {{E_1}}}} \right) = {3 \over 4} \Rightarrow {{P({E_2} \cap {E_1})} \over {P({E_1})}} = {3 \over 4}\]
\[P({E_1} \cap {E_2}) = {1 \over 8}\]
\[P({E_2}) = {1 \over 4},\,P({E_1}) = {1 \over 6}\]
(A) \[P({E_1} \cap {E_2}) = {1 \over 8}\] and \[P({E_1})\,.\,P({E_2}) = {1 \over {24}}\]
\[ \Rightarrow P({E_1} \cap {E_2}) \ne P({E_1})\,.\,P({E_2})\]
(B) \[P(E{'_1} \cap E{'_2}) = 1 - P({E_1} \cup {E_2})\]
\[ = 1 - \left[ {{1 \over 4} + {1 \over 6} - {1 \over 8}} \right] = {{17} \over {24}}\]
\[P(E{'_1}) = {3 \over 4} \Rightarrow P(E{'_1})P({E_2}) = {3 \over {24}}\]
\[ \Rightarrow P(E{'_1} \cap E{'_2}) \ne P(E{'_1})\,.\,P({E_2})\]
(C) \[P({E_1} \cap E{'_2}) = P({E_1}) - P({E_1} \cap {E_2})\]
\[ = {1 \over 6} - {1 \over 8} = {1 \over {24}}\]
\[P({E_1})\,.\,P({E_2}) = {1 \over {24}}\]
\[ \Rightarrow P({E_1} \cap E{'_2}) = P({E_1})\,.\,P({E_2})\]
(D) \[P(E{'_1} \cap {E_2}) = P({E_2}) - P({E_1} \cap {E_2})\]
\[ = {1 \over 4} - {1 \over 8} = {1 \over 8}\]
\[P({E_1})P({E_2}) = {1 \over {24}}\]
\[ \Rightarrow P(E{'_1} \cap {E_2}) \ne P({E_1})\,.\,P({E_2})\]