Tree Diagrams and the Multiplication Rule

A tree diagram breaks a multi-stage chance process into stages, with each stage splitting into branches for its possible outcomes. The first set of branches comes from the stage one event, and from the end of each of those, a new set of branches shows what can happen at stage two. Once the tree is drawn, you don't have to think hard about the probability rules anymore: you just follow a path from start to finish and multiply the numbers written on the branches you passed through. That multiplication works because of the general multiplication rule: P(A and B) = P(A) x P(B given A). Each branch already accounts for what happened before it, so multiplying along the path automatically applies that conditioning correctly.

The branching matters most when draws are without replacement, because the second-stage probabilities change depending on which branch you took first. Drawing a red marble first leaves fewer red marbles and fewer total marbles behind, so the branch probabilities for the second draw are different on the red path than on the blue path. The tree keeps these cases separate so you never apply the wrong denominator.

This topic is the bridge between the basic probability rules from earlier in the unit and the more compact formulas coming next, like the binomial setting. Once events become independent and repeated many times, the tree gets too wide to draw, and the binomial formula takes over the same multiplying-along-a-path idea in a shortcut form.