The rule for conditional probability gives that
$P (A\mid B)P (B)=P (B\mid A)P (A)=P (A \cap B) $
When I am considering the first two expressions, I am confused. Why should they be equal? If event B occurring changes the probability of event A occurring, then I just don't understand why they would necessarily change in such a way that the probability of A occurring after B (multiplied by the probability of B) would necessarily be the same as the probability of B occurring after A (multiplied by P (A)).
I just cannot understand how the changes in probabilities of A and B must be related in that way.
Any help in understanding this (examples and developing intuition would be great) would be much appreciated. I really want to develop a better intuitive understanding of why some mathematical statements must be true, not just by churning through the mathematics and seeing that it leads to this result.
Additional note: I have tried to think about this in terms of Venn diagrams, however I do not see how this can tell us anything. When I consider a Venn diagram, I think of it showing the probabilities of A and B occurring at any given point in time. Now of B occurs, then the probability of A occurring would change too. So surely the Venn diagram would change and the Venn diagram drawn before does not tell us anything about the situation now?