A probability tree diagram shows all possible outcomes of two or more events and their probabilities in a structured way.
Rules:
- Probabilities along each set of branches must sum to 1
- To find the probability of a combined outcome, multiply along the branches
- To find the probability of multiple routes giving the same outcome, add those products
Example: A biased coin: $P(H) = 0.6$, $P(T) = 0.4$. Flipped twice. $$P(HH) = 0.6 \times 0.6 = 0.36$$ $$P(\text{exactly one head}) = P(HT) + P(TH) = 0.6 \times 0.4 + 0.4 \times 0.6 = 0.48$$
Memory aid: Multiply along branches, add down the list.
When events are dependent (e.g. without replacement), adjust the probabilities on the second set of branches based on the first outcome.
Common error: adding instead of multiplying along branches, or multiplying all outcomes when you should only add the desired ones.