Independent events

**matty888** · April 29th 2008, 05:31 AM

If the event E is independent of the event F,show that
i)

ii)

**Plato** · April 29th 2008, 08:24 AM

By independence we know that $P(E \cap F) = P(E)P(F)$ .
$\begin{array}{l} P(E) = P(E \cap F) + P(E \cap \overline F \,) \\ <br /> P(E \cap \overline F ) = P(E) - P(E \cap F) \\ P(E \cap \overline F ) = P(E)\left[ {1 - P(F)} \right] = P(E)P\left( {\overline F } \right) \\ \end{array}$

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.

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