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Tree diagrams and conditional probability explained

Draw tree diagrams for independent and dependent events, including picks without replacement, with worked examples and common mistakes.

IntermediateUpdated 5 Nov 2026
GCSEHigh School

Curriculum fit: GCSE Maths Probability (Foundation and Higher; conditional probability is mainly Higher). US: High School Statistics and Probability (Conditional Probability and the Rules of Probability).

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What is it?

A tree diagram shows all the possible outcomes of two or more events. Each branch is labelled with its probability. To find the probability of a path, multiply along the branches. To find the probability of several paths, add them.

Events are independent if one does not affect the other. They are dependent if the first changes the probabilities of the second, such as picking sweets without replacement.

Key formulas and ideas

  • Independent events: P(A and B) = P(A) × P(B).
  • Dependent events: P(A and B) = P(A) × P(B given A).
  • Add probabilities for 'or' between different paths.
  • P(at least one) = 1 − P(none).

Worked examples

Example 1: Without replacement

Question: A bag has 3 red and 2 blue counters. Two are taken without replacement. Find P(both red).

  1. First red: 3/5.
  2. Second red, given first red: 2/4.
  3. Multiply: 3/5 × 2/4 = 6/20 = 3/10.

Answer: 3/10

Example 2: One of each colour

Question: Using the same bag, find P(one red and one blue).

  1. Red then blue: 3/5 × 2/4 = 6/20.
  2. Blue then red: 2/5 × 3/4 = 6/20.
  3. Add: 12/20 = 3/5.

Answer: 3/5

Example 3: At least one

Question: A fair coin is tossed twice. Find P(at least one head).

  1. P(no heads) = 1/2 × 1/2 = 1/4.
  2. P(at least one head) = 1 − 1/4 = 3/4.

Answer: 3/4

Common mistakes

  • Adding along the branches instead of multiplying.
  • Not changing the second set of probabilities when there is no replacement.
  • Forgetting to add together all the paths that match.
  • Branch probabilities that do not sum to 1.

Practice questions

1. P(head) = 1/2 and P(6 on a die) = 1/6. P(head and 6)?
2. A bag has 4 red and 6 blue. Two are taken without replacement. P(both blue)?
3. For the same bag, P(at least one red)?
4. P(rain) = 0.3 and P(late given rain) = 0.5. P(rain and late)?
5. On a tree diagram, you multiply probabilities…

Frequently asked questions

How can I check a tree diagram?

The branches from each point should add up to 1, and the probabilities of all the final outcomes should add up to 1.

What does 'conditional probability' mean?

The probability of an event given that another event has already happened.

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Sources and review

Reviewed by StudyMorrow editorial review on 5 Nov 2026.