# Independent events

• Apr 29th 2008, 05:31 AM
matty888
Independent events
• Apr 29th 2008, 08:24 AM
Plato
By independence we know that $P(E \cap F) = P(E)P(F)$.
$\begin{array}{l} P(E) = P(E \cap F) + P(E \cap \overline F \,) \\
P(E \cap \overline F ) = P(E) - P(E \cap F) \\ P(E \cap \overline F ) = P(E)\left[ {1 - P(F)} \right] = P(E)P\left( {\overline F } \right) \\ \end{array}$

For the second part just switch roles.
Because $\overline F \,\& \,E$ independent by the above we know that $\overline F \,\& \,\overline E$ must be also.