Venn Diagram vs Tree Diagram

Each circle holds the outcomes in one event, the lens where circles overlap is the intersection $A \cap B$, and the area outside every circle is the complement of the union. Filling in the overlap first keeps you from double counting, which is the same idea the addition rule enforces with subtraction. In a class of 30 students, 18 play a sport and 12 play an instrument with 7 doing both, so the sport-only region holds $18 - 7 = 11$, the instrument-only region holds $12 - 7 = 5$, and $30 - (11 + 7 + 5) = 7$ students do neither. Mutually exclusive events show up as circles that do not touch.

Branches at the first stage carry the probabilities of the starting events, and branches further out carry conditional probabilities given the path so far. Multiplying along a path is just the general multiplication rule $P(A \cap B) = P(A) \cdot P(B \mid A)$ drawn in picture form. From a bag of 3 red and 2 blue chips, drawing two without replacement gives $P(\text{red then blue}) = \frac{3}{5} \cdot \frac{2}{4} = 0.3$. To get the probability of a result that several paths lead to, add those path probabilities.