Why is the denominator in a conditional probability the probability of the conditioning event? Quite a simple or at least short question: Why is $ \frac{P(A \cap B)}{P(B)} $ divided by $ P(B) $ for the conditional probability?
$ P(A | B) = \frac{P(A \cap B)}{P(B)} $
Random image to visualize: 
I actually would like to make use out of the tree display which I can't grasp at all. Why would I divide by $ P(A+) $ when I had to move along this path?
Sorry for the mix-up of $A+$ and $ B $..


First of all, I disagree with the picture. Blue set does not refer to $P(A)$ but $P(A\cap B')$. Green set refers to $P(B\cap A')$.
Back to the question: Why
$$
P(B^+|A^+)=\frac{P(B^+\cap A^+)}{P(A^+)}?
$$
We can understand it by saying $P(B^+|A^+)$ as the probability that event $B^+$ happening given that event $A^+$ happened.
It means that, if I you have information that $A^+$ has happened,

*

*you must make sure that both events happened at the same time. Hence, you have $P(B^+\cap A^+)$.

*your answers must be with respect to the event $A^+$. Your answers are constrained to event $A^+$. Hence, you have a ratio to $P(A^+)$.

As an example, let $A^+$ be the event someone goes outside to play and $B^+$ be the event that someone will come home after 6pm. If you know that someone has already went outside to play, what is their probability to come home after 6pm? You already know that $A^+$ happened, so you will have

*

*that person went outside to play AND that person will come home after 6pm.

*that the probability above happened with respect to that person will come home after 6pm.

