Bayes Formula: Intuition I found an intuitive explanation for Bayes' Theorem but I do not understand point 3:

With $P(A|B) = \frac{P(A \cap B)}{P(B)}$ I see that P(B) is a scaling factor for partitions of B to be able to sum up to 1. But I can't seem to wrap my head around the meaning of the ratio in Bayes' Theorem.

I can give you a similar interpretation. Write it as
$$\Pr(B|A)=\frac{\Pr(A|B)\Pr(B)}{\Pr(A)}=\frac{\Pr(B\cap A)}{\Pr(A)}$$ 
First notice the difference between the two events $B$ and $B|A$. The former is an event in the sample space $\Omega$ whose probability is one. We can write $\Pr(B)$ as $$\frac{\Pr(B\cap\Omega)}{\Pr(\Omega)}=\frac{\Pr(B)}{\Pr(\Omega)}=\frac{\Pr(B)}{1}$$
However, $B|A$ means that for event $B$ the sample space is reduced from $\Omega$ to $A$, hence we have $\Pr(A)$ in denomerator instead of $\Pr(\Omega)$. Similarly, in the numerator we have $\Pr(B\cap A)$ since we only consider $B$ in a subset of the sample space that intersects with $A$.
