If the event E is independent of the event F,show that
i)
ii)
By independence we know that $\displaystyle P(E \cap F) = P(E)P(F)$.
$\displaystyle \begin{array}{l} P(E) = P(E \cap F) + P(E \cap \overline F \,) \\
P(E \cap \overline F ) = P(E) - P(E \cap F) \\ P(E \cap \overline F ) = P(E)\left[ {1 - P(F)} \right] = P(E)P\left( {\overline F } \right) \\ \end{array}$
For the second part just switch roles.
Because $\displaystyle \overline F \,\& \,E$ independent by the above we know that $\displaystyle \overline F \,\& \,\overline E $ must be also.