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Tree diagrams questions and answers

A probability tree diagram shows two or more events in a row. Multiply along the branches to get the probability of a path; add the paths you want. Without replacement, the second-stage probabilities change because there's one fewer item. For "at least one", it's often quickest to work out 1−P(none)1 - P(\text{none}).

8 questions · Foundation tier · Every answer is checked as you type it, with a full worked solution. Updated .

Start the questions Tree diagrams guide

Where tree diagrams comes up

GCSE (AQA, Edexcel, OCR)Foundation and Higher: tree diagrams for independent and dependent combined events. Higher only: conditional probability from tree diagrams.
Cambridge IGCSE (0580)Core: tree diagrams with replacement only. Extended: with or without replacement, and conditional probability.

The method

  1. Each set of branches adds to 1.
  2. Multiply along a path.
  3. Add the paths that give the outcome you want.
  4. Without replacement: reduce the numbers on the second branches.
  5. At least one: 1 − P(none).

Tree diagrams questions

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Warm-up

Question 1Warm-upFoundation

The probability it rains on any day is 0.3, independently. Find the probability it rains on both Monday and Tuesday.

Question 2Warm-upFoundation

Using the same probabilities, find the probability it rains on neither day.

Exam-style

Question 3Exam-styleFoundation

Find the probability it rains on exactly one of the two days.

Question 4Exam-styleFoundation

A bag has 4 red and 6 blue counters. One is taken, replaced, then another taken. Find P(both red).

Question 5Exam-styleFoundation

The same bag, but the first counter is not replaced. Find P(both red).

Question 6Exam-styleFoundation

Without replacement again, find P(one red and one blue).

Challenge

Question 7ChallengeFoundation

With replacement, find P(at least one red) from the bag of 4 red and 6 blue.

Question 8ChallengeFoundation

P(rain) = 0.4. If it rains, P(Sam is late) = 0.2; if not, P(Sam is late) = 0.05. Find P(Sam is late).

Common mistakes

  • Adding along branches instead of multiplying.
  • Forgetting there are two orders (red then blue, blue then red).
  • Not changing the denominator without replacement.

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Sources

The questions are our own. The “where it comes up” notes are from each board's published specification, checked on 26 October 2026:

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