Sorry to say, I cannot follow what you have done.
But the proof is simple.
It is well known that .
So .
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)