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Pr(A given or │ B)= Pr(A intersection B)/Pr(B)

The the proof for it is self evident when one multiplies both sides by Pr(B).

However, proofs are no use to me if I do not have a feel for why it works. To rid my question of ambiguity: If a mathematician is doodling on some paper and gets: Pr(A intersection B)=(Pr(A intersection B)*Pr(B))/Pr(B) then changes it into Pr(A│B)=Pr(A intersection B)/Pr(B) how will he/she know that Pr(A│B) means probability of A given B? How will they know that this formula pertains to something?

I fear my question may still be steeped in ambiguity...sorry if it is.