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Thread: 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 $\displaystyle P(E \cap F) = P(E)P(F)$.
    $\displaystyle \begin{array}{l} P(E) = P(E \cap F) + P(E \cap \overline F \,) \\
    P(E \cap \overline F ) = P(E) - P(E \cap F) \\ P(E \cap \overline F ) = P(E)\left[ {1 - P(F)} \right] = P(E)P\left( {\overline F } \right) \\ \end{array}$

    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.
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