Two bags, coloured marbles, draw one from each. Trying to track the outcomes in your head is hopeless. Draw a tree, and every path is a possibility — multiply along, add across.
A probability tree diagram maps a multi-stage experiment: each branch is an outcome, labelled with its probability. Follow a path and multiply; combine paths and add.
Multi-step games, medical test sequences, conditional probability, risk chains — trees turn tangled 'what if' problems into something you can read off a page.
Draw a branch for each outcome of the first event. Write its probability on the branch. The probabilities on branches from one point must sum to 1.
Flip a coin, then roll a die. What's P(heads AND a six)?
A bag has 3 red, 2 blue. Draw two without replacement. P(both red)?
First red: 3/5. Then red again (one fewer red, one fewer total): 2/4. Multiply: 3/5 × 2/4 = 6/20 = 3/10.
With 'without replacement', the second branch's probabilities change. Forgetting to adjust (using 3/5 again instead of 2/4) is the most common tree error.
Sanity check: probabilities on branches from any single point must add to 1. If they don't, you've missed an outcome or mislabelled one.
- Each branch = an outcome with its probability; forks sum to 1.
- Multiply along a path; add across paths that satisfy your event.
- 'Without replacement' changes the later branch probabilities.