Conditional Probability

Sometimes you already know that one thing has happened, and you want the chance of another. The conditional probability of B given that A has happened is written P(B \mid A).

Knowing that A happened shrinks the sample space to just the A outcomes — so out of those, you ask how many are also B:

P(B \mid A) = \frac{P(A \text{ and } B)}{P(A)}

On a tree diagram, the second-stage branches are exactly these conditional probabilities — each one assumes the first-stage outcome has already happened.

The chance of B given A is P(B \mid A) = \dfrac{P(A \text{ and } B)}{P(A)}.
The word "given" reduces the sample space to just the A cases — you only count outcomes inside A.
For independent events, P(B \mid A) = P(B) — knowing A changes nothing.

See it: given A, look only inside A

Here A has 6 + 4 = 10 people in it. Once you are told the person is in A, ignore everyone else — the chance they are also in B is the 4 overlap out of those 10.