
probability proof help
Prove that P(E) = P(X)P(EX) + P(X*)P(EX*)
My question is not about the actual proof but about the "=". I have to prove both P(E)>P(X)P(EX) and P(X)P(EX)> P(E)?
I can prove P(X)P(EX)> P(E) but how do I prove P(E)> P(X)(P(EX)?
Is this a valid proof?
P(EX)P)(X) = [P(E intersection X)/P(X)]x[(P(X)] + [P(E intersection X*)/P(X*)]x[(P(X*)] = P(E)

Sorry to say, I cannot follow what you have done.
But the proof is simple.
It is well known that $\displaystyle P(EX) = \frac{{P(E \cap X)}}
{{P(X)}}\;\& \;P(E) = P(E \cap X) + P(E \cap X^* )$.
So $\displaystyle P(E) = P(E \cap X) + P(E \cap X^* ) = P(EX)P(X) + P(EX^* )P(X^* )$.