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Math Help - probability proof help

  1. #1
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    probability proof help

    Prove that P(E) = P(X)P(E|X) + P(X*)P(E|X*)

    My question is not about the actual proof but about the "=". I have to prove both P(E)->P(X)P(E|X) and P(X)P(E|X)-> P(E)?

    I can prove P(X)P(E|X)-> P(E) but how do I prove P(E)-> P(X)(P(E|X)?

    Is this a valid proof?

    P(E|X)P)(X) = [P(E intersection X)/P(X)]x[(P(X)] + [P(E intersection X*)/P(X*)]x[(P(X*)] = P(E)
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  2. #2
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    Sorry to say, I cannot follow what you have done.
    But the proof is simple.
    It is well known that P(E|X) = \frac{{P(E \cap X)}}<br />
{{P(X)}}\;\& \;P(E) = P(E \cap X) + P(E \cap X^* ).
    So P(E) = P(E \cap X) + P(E \cap X^* ) = P(E|X)P(X) + P(E|X^* )P(X^* ).
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