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Math Help - Independent events

  1. #1
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    Independent events

    If the event E is independent of the event F,show that
    i)
    ii)
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  2. #2
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    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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