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. 

It's equivalent to $$\frac{n(A\cap{B})}{n(B)}$$ in other words the proportion of the members of set $B$ which are also members of set $A$.
