Probability tree diagrams organize multi-step chance events by showing each possible outcome as a branch. They are useful because many real situations happen in stages, such as drawing counters, tossing coins, or choosing cards. A tree makes it easier to see all possible paths and avoid missing outcomes.
It also connects naturally to the basic rules for multiplying and adding probabilities.
Key Facts
- For a path through a tree, multiply branch probabilities: P(A and B) = P(A) × P(B given A).
- To combine separate paths that lead to the same final event, add their probabilities: P(path 1 or path 2) = P(path 1) + P(path 2).
- The probabilities on all branches leaving the same node must add to 1.
- With replacement, probabilities usually stay the same from one stage to the next.
- Without replacement, probabilities often change because the total number of items and the number of favorable items change.
- For independent events, P(B given A) = P(B), so P(A and B) = P(A) × P(B).
Vocabulary
- Probability tree diagram
- A branching diagram that shows all possible outcomes of a multi-stage probability experiment.
- Branch
- A line in a probability tree that represents one possible outcome at a stage.
- Path
- A complete route from the start of a tree to a final outcome.
- With replacement
- A situation where an item is put back before the next selection, so the probabilities may stay the same.
- Without replacement
- A situation where an item is not put back before the next selection, so later probabilities may change.
Common Mistakes to Avoid
- Adding along a single path is wrong because a path represents events happening together. Multiply the branch probabilities to find the probability of that exact sequence.
- Forgetting to change probabilities without replacement is wrong because the contents of the group have changed after the first selection. Update both the favorable count and the total count.
- Adding probabilities from overlapping paths is wrong because it can count the same outcome more than once. Only add separate, mutually exclusive paths unless you correct for overlap.
- Leaving branch probabilities that do not add to 1 is wrong because each set of branches from one node should include all possible outcomes at that stage. Check each split before calculating final probabilities.
Practice Questions
- 1 A bag has 3 red counters and 2 blue counters. One counter is drawn, replaced, and a second counter is drawn. Use a probability tree to find P(red then blue).
- 2 A bag has 4 green counters and 3 yellow counters. Two counters are drawn without replacement. Use a probability tree to find P(two yellow counters).
- 3 A student says that replacement does not matter in a probability tree because the same colors are still possible on the second draw. Explain why this reasoning is incomplete.