Proving Independence of A & B: Probability Proof

[jaejoon89]

Homework Statement

Prove if P(A|B) = P(A|B') then A and B are independent.

where B' is the complement of B

Homework Equations

if independent, P(A|B) = P(A)
also, P(A∩B) = P(A)P(B)

for conditional probability,
P(A|B) = P(A∩B) / P(B)

The Attempt at a Solution

P(A|B) = P(A∩B) / P(B) = P(B|A)P(A) / P(B)
P(A|B') = P(A∩B') / P(B') = P(B'|A)P(A) / P(B')

I'm not really sure how to go from here... What do I do?

 

[Elucidus]

If you know that P(A|B) = P(A|B') and

P(A|B) = P(AB)/P(B) and P(A|B') = P(AB')/P(B')

(where AB = A intersect B)

then why not set these right-hand sides equal and see what happens?

--Elucidus

 

[jaejoon89]

Yeah, I did that. Then P(A∩B) / P(B) = P(A∩B') / P(B') where ∩ means intersection

Sorry, I still don't see what should follow.