Intuition behid $P(A\mid B)$. What is the intuition behind the formula $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$
I have seen this formula around, but every site/book I look at does not really have a clear & cut explanation behind this formula. 

All probabilities are the number of desired outcomes divided by the total possible outcomes. 
So P(A|B) is the number ways A and B occur divided by the number B can occur.
In other words: #A&B / #B.
This is equal to (#A&B / #everything)(#everything / #B)
Which is the RHS.
Although I prefer to think of it simply that the LHS refers to probability of A given the limited range that B occurs as well, which reasonably should be the probability that A and B both occurring in the "big" world but then reduced to the requirement that B occured.  Which is pretty intuitive to me.
