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. 
 A: Imagine a venn diagram, with two circles, labelled $A$ and $B$. The quantity within each shaded region represents a Probability of an Event. Say, if you were to shade the $A$ entirely then this would give you $P(A)$. Now,let the common region be denoted by $P(A \cap B)$. Considering your question, you want the probability $P(A|B)$, which is the probability of $A$ occurring given that $B$ has already happened. Now think about it in terms of the Venn Diagram. The only region where $B$ and $A$ events can occur is in the overlapping region, therefore you get a numerator $P(A \cap B)$ as your favorable outcome. But this time, the total number of outcome is all of $A$ therefore $P(A)$ is in the denominator.   
A: 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 $A$.
A: 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.
