I have a question about a nagging issue I have in probability -
The conditional probability can be expressed thus: p(A|B)=p(AB)/p(B) However, the proofs I've seen of this rely on restricting your initial sample space to B. Doesn't this limit the use of this equivalency to cases where you are, in fact, conditioning on B - that is, you can't use this to make inferences about B's conditional probability given A? Or am I misunderstanding the proof? (Or is there another proof I haven't seen?)
(I can't think of a case where you can't make inferences about B given A, but I'm having trouble ascertaining whether the proof actually holds.)
Can you link to such a proof?
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