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MathidiaMathematics Academy
Probability foundations
Lesson 48 of 75
Standard practice

Probability trees

Objective. Model sequential events with changing probabilities.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Probability trees model sequences of events by branching, where each branch is labelled with its conditional probability given the path taken to reach it, and probabilities along a single path are multiplied together to find the probability of that complete sequence. To find the probability of a particular final outcome, sum the probabilities of all paths leading to that outcome. For example, drawing two balls without replacement from a bag, the second branch's probabilities change depending on what was drawn first, since one ball has been removed. Tree diagrams are especially useful for problems involving several dependent stages where probabilities update between stages.

Worked example

A bag contains 5 red and 3 blue balls. Two balls are drawn without replacement. Use a probability tree to find the probability both balls are red.

5 … 3
Illustrated card showing 5 … 3
  1. 1First branch: P(red first) = 5/8.
  2. 2Second branch given first was red: P(red second|red first) = 4/7, since one red ball and one total ball are removed.
  3. 3Multiply along the path: P(both red) = (5/8)×(4/7).
  4. 4Simplify: (5×4)/(8×7) = 20/56 = 5/14.

Answer: 5/14

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Practice

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Common mistake. Students often reuse the original probabilities on later branches for sampling without replacement, forgetting that removing an item changes the remaining totals. Always recalculate probabilities for later branches based on what remains after each stage.

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