Tree Diagrams

Many experiments happen in stages — flip a coin, then flip it again; or check the weather today, then tomorrow. A tree diagram draws each stage as a set of branches, with the probability written on every branch. Following one path from the start to an endpoint traces one complete outcome.

Two rules turn the picture into numbers:

Multiply ALONG the branches of a path — that gives the probability of this and then that. For the top path, 0.6 \times 0.7 = 0.42.
Add the path-probabilities for the different ways an event can happen — this way or that way.

And a built-in check: the branches at each split add up to 1 (0.6 + 0.4 = 1), and so do all four endpoint probabilities (0.42 + 0.18 + 0.08 + 0.32 = 1).

For a multi-stage experiment drawn as a tree:

the branch probabilities coming out of each split sum to 1 (they cover every possibility at that stage);
multiply ALONG a path (AND) to get the probability of that whole sequence of outcomes;
add BETWEEN paths (OR) to combine the separate ways an event can occur;
this works whether or not the stages are independent — if a later split depends on the earlier outcome, just read that split's branch probabilities off its own branches.