Tree diagrams

Branched diagrams for two-stage events; multiply along branches, add across pathways.

Each branch shows an outcome with its probability. Probabilities along each path multiply. Add path probabilities for ‘OR’ combined outcomes. Without replacement: second-stage probabilities depend on first-stage outcome.

Worked examples

Two coin flips: P(2 heads) = ½ × ½ = ¼.

P(at least one head) = 1 − P(both tails) = 1 − ¼ = ¾.

Bag of 5 red, 3 blue, no replacement: P(red, then red) = (5/8) × (4/7).

Frequently asked questions

‘At least one’?

Easier as 1 − P(none). Avoids enumerating multiple branches.

All probabilities sum to 1?

Down the entire tree, all final-leaf probabilities sum to 1.