# probability proof help

• September 17th 2008, 07:23 AM
lord12
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)
• September 17th 2008, 08:39 AM
Plato
Sorry to say, I cannot follow what you have done.
But the proof is simple.
It is well known that $P(E|X) = \frac{{P(E \cap X)}}
{{P(X)}}\;\& \;P(E) = P(E \cap X) + P(E \cap X^* )$
.
So $P(E) = P(E \cap X) + P(E \cap X^* ) = P(E|X)P(X) + P(E|X^* )P(X^* )$.