Conditional tree diagrams
Objective. Complete trees from dependent conditional information.
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- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Conditional tree diagrams show sequences of dependent events where the probability of a later branch changes depending on what happened earlier, unlike independent events with fixed probabilities. Each branch after the first is labelled with a conditional probability, and probabilities along any path are multiplied together to find the probability of that whole sequence. Branches from the same point must always sum to 1. For drawing two balls without replacement from a bag of 5 red and 3 blue, the second draw's probabilities depend on the colour removed first. Completing a tree means filling every missing branch using this sum-to-1 rule before multiplying along paths.
Worked example
A bag has 4 red and 6 blue counters. Two counters are drawn without replacement. Using a tree diagram, find the probability both are red.
- 1P(first red) = 4/10
- 2Given first red, P(second red) = 3/9
- 3Multiply along the branch: (4/10) × (3/9)
- 4Probability = 12/90 = 2/15
Answer: 2/15
Concept mastery
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Practice
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Start practisingCommon mistake. Students often reuse the first event's probabilities for the second branch even though an item was removed without replacement, which ignores the changed total and gives incorrect conditional values.
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