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GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 2–7 · Probability

Frequency Trees and Mutually Exclusive Events

A frequency tree splits a group into categories, then splits each category again. It is a GCSE-specific diagram that IGCSE textbooks skip, and a regular Foundation question. This page also covers the rule that probabilities must add up to 1.

1Frequency trees

A frequency tree shows actual numbers, not probabilities. Each pair of branches must add up to the number they came from.

Students 80Boys 35Passed 21Failed 14Girls 45Passed 27Failed 18
80 students take a driving test: 35 boys (21 passed) and 45 girls (27 passed)
Key ideaThe numbers on each pair of branches add to the number before: 35 + 45 = 80; 21 + 14 = 35; 27 + 18 = 45.

2Completing a frequency tree

Worked example
80 students took a test. 35 were boys. 21 of the boys passed. 18 girls failed. Complete the frequency tree.
  1. Girls = 80 − 35 = 45.
  2. Boys who failed = 35 − 21 = 14.
  3. Girls who passed = 45 − 18 = 27.
See the tree above

3Probability from a frequency tree

Worked example
Using the tree above, a student is chosen at random. Find (a) P(passed) (b) P(girl who passed) (c) the probability that a student who passed is a boy.
  1. (a) Passed = 21 + 27 = 48, so P = 4880 = 35.
  2. (b) 2780.
  3. (c) Higher: there are 48 students who passed, and 21 of them are boys: 2148 = 716.
(a) 3/5 (b) 27/80 (c) 7/16
Exam tipPart (c) is a conditional probability. The denominator is only the students who passed. This is a common grade 6–7 question.

4Probabilities add up to 1

If a list of outcomes covers everything that can happen (the outcomes are exhaustive) and no two can happen together (they are mutually exclusive), their probabilities add up to 1.

Worked example
A spinner lands on red, blue, green or yellow.
ColourRedBlueGreenYellow
Probability0.150.20.35x
Find x.
  1. 0.15 + 0.2 + 0.35 = 0.7.
  2. x = 1 − 0.7.
x = 0.3
ComplementP(not A) = 1 − P(A)

5The OR rule for mutually exclusive events

Addition ruleIf A and B can’t happen at the same time: P(A or B) = P(A) + P(B)
Worked example
Using the spinner above, find P(red or green).
  1. Red and green can’t happen together, so add: 0.15 + 0.35.
0.5
Only for mutually exclusive events
  • Using the addition rule for events that can happen together, such as “red” and “a prime number” on a dice. Use a Venn diagram or a list for those.

6Expected frequency

Expected number of times = probability × number of trials.

Worked example
The spinner is spun 200 times. How many times would you expect green?
  1. 0.35 × 200.
70 times
Common mistakes
  • Writing probabilities on a frequency tree. It shows numbers.
  • Pairs of branches that don’t add up to the number they came from.
  • Using the grand total as the denominator for a conditional (“given that”) probability.
  • Adding probabilities of events that are not mutually exclusive.
  • Forgetting that the probabilities of all possible outcomes must total 1.

★Key Vocabulary Recap

Frequency tree
A diagram that splits a total into categories using actual counts.
Mutually exclusive
Events that can’t happen at the same time.
Exhaustive
A set of outcomes that covers every possibility.
Expected frequency
Probability × number of trials.
Conditional probability
A probability given that something else has happened.
Specification reference (Pearson Edexcel GCSE (9–1) Mathematics 1MA1: P1, P3, P4, P9)
  • P1: record, describe and analyse the frequency of outcomes of probability experiments using tables and frequency trees.
  • P3: relate relative expected frequencies to theoretical probability.
  • P4: apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one.
  • P9 (Higher): calculate and interpret conditional probabilities using expected frequencies with frequency trees.
Next stepReady to practise? Try the 10 exam-style questions, then check your working against the worked solutions.
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Practice questions

GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 2–7 · Probability

Frequency Trees and Mutually Exclusive Events — Practice Questions

10 exam-style questions (27 marks), building from warm-up to grade 8–9. Show full working: Edexcel gives method marks even when the final answer is wrong.

Foundation · Grades 2–3
  1. (2 marks)
    Grade 2
    A spinner can land on A, B or C. P(A) = 0.3 and P(B) = 0.4. Find P(C).
  2. (3 marks)
    Grade 3
    60 people chose between cinema and bowling. 24 were adults. 9 adults chose cinema. 22 children chose bowling. Complete the frequency tree.
    People 60Adults 24Cinema 9BowlingChildrenCinemaBowling 22
  3. (2 marks)
    Grade 3
    Using your tree from Question 2, a person is chosen at random. Find the probability that they chose cinema.
Foundation and Higher · Grades 4–5
  1. (2 marks)
    Grade 4
    A bag contains red, blue and green counters. P(red) = 14 and P(blue) = 13. Find P(green).
  2. (3 marks)
    Grade 4
    A spinner is biased.
    ColourRedBlueGreenYellow
    Probability0.150.20.35x
    (a) Find x. (b) Find P(red or blue). (c) The spinner is spun 200 times. How many times would you expect green?
  3. (3 marks)
    Grade 5
    A dice is biased. The probabilities of 4, 5 and 6 are equal. P(1) = 0.1, P(2) = 0.2, P(3) = 0.4. Find P(6).
  4. (4 marks)
    Grade 5
    120 patients had a test. 48 of them have a condition. Of those with the condition, 30 tested positive. Of those without it, 60 tested negative.
    (a) Draw a frequency tree. (b) How many patients tested positive?
Higher · Grades 6–7
  1. (2 marks)
    Grade 6
    Using Question 7, a patient with the condition is chosen at random. Find the probability that they tested positive.
  2. (3 marks)
    Grade 7
    Using your tree from Question 2, a person who chose cinema is selected at random. Find the probability that they are a child.
  3. (3 marks)
    Grade 7
    In a school, 36 students study Art. 14 of the Art students also study Music. One Art student is chosen at random. Find the probability that they also study Music, and explain why the events “studies Art” and “studies Music” are not mutually exclusive.
When you’re doneRevise the method in the notes, then mark your answers with the worked solutions.
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Worked solutions

Show worked solutions
GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 2–7 · Probability

Frequency Trees and Mutually Exclusive Events — Worked Solutions

Full worked solutions to all 10 questions (27 marks), set out the way an Edexcel examiner wants to see them.

Foundation · Grades 2–3
  1. (2 marks)
    Grade 2
    A spinner can land on A, B or C. P(A) = 0.3 and P(B) = 0.4. Find P(C).
    Solution
    1. Probabilities add to 1: 1 − 0.3 − 0.4.
    0.3
  2. (3 marks)
    Grade 3
    60 people chose between cinema and bowling. 24 were adults. 9 adults chose cinema. 22 children chose bowling. Complete the frequency tree.
    People 60Adults 24Cinema 9BowlingChildrenCinemaBowling 22
    Solution
    1. Adults bowling = 24 − 9 = 15.
    2. Children = 60 − 24 = 36.
    3. Children cinema = 36 − 22 = 14.
    People 60Adults 24Cinema 9Bowling 15Children 36Cinema 14Bowling 22
    15, 36, 14
  3. (2 marks)
    Grade 3
    Using your tree from Question 2, a person is chosen at random. Find the probability that they chose cinema.
    Solution
    1. Cinema = 9 + 14 = 23.
    2. P = 2360.
    2360
Foundation and Higher · Grades 4–5
  1. (2 marks)
    Grade 4
    A bag contains red, blue and green counters. P(red) = 14 and P(blue) = 13. Find P(green).
    Solution
    1. 1 − 14 − 13 = 1212 − 312 − 412.
    512
  2. (3 marks)
    Grade 4
    A spinner is biased.
    ColourRedBlueGreenYellow
    Probability0.150.20.35x
    (a) Find x. (b) Find P(red or blue). (c) The spinner is spun 200 times. How many times would you expect green?
    Solution
    1. (a) x = 1 − 0.7 = 0.3.
    2. (b) 0.15 + 0.2 = 0.35.
    3. (c) 0.35 × 200 = 70.
    (a) 0.3 (b) 0.35 (c) 70
  3. (3 marks)
    Grade 5
    A dice is biased. The probabilities of 4, 5 and 6 are equal. P(1) = 0.1, P(2) = 0.2, P(3) = 0.4. Find P(6).
    Solution
    1. P(4) + P(5) + P(6) = 1 − 0.7 = 0.3.
    2. They are equal, so each is 0.3 ÷ 3.
    0.1
  4. (4 marks)
    Grade 5
    120 patients had a test. 48 of them have a condition. Of those with the condition, 30 tested positive. Of those without it, 60 tested negative.
    (a) Draw a frequency tree. (b) How many patients tested positive?
    Solution
    1. Without the condition: 120 − 48 = 72. With it: 30 positive, 48 − 30 = 18 negative.
    2. Without it: 60 negative, so 72 − 60 = 12 positive.
    3. (b) Positive = 30 + 12 = 42.
    Patients 120Condition 48Positive 30Negative 18No condition 72Positive 12Negative 60
    (b) 42
Higher · Grades 6–7
  1. (2 marks)
    Grade 6
    Using Question 7, a patient with the condition is chosen at random. Find the probability that they tested positive.
    Solution
    1. Choose only from the 48 with the condition: 30 tested positive.
    2. 3048 = 58.
    58
  2. (3 marks)
    Grade 7
    Using your tree from Question 2, a person who chose cinema is selected at random. Find the probability that they are a child.
    Solution
    1. Cinema total = 9 + 14 = 23.
    2. Children among them = 14.
    3. P = 1423.
    1423
  3. (3 marks)
    Grade 7
    In a school, 36 students study Art. 14 of the Art students also study Music. One Art student is chosen at random. Find the probability that they also study Music, and explain why the events “studies Art” and “studies Music” are not mutually exclusive.
    Solution
    1. P = 1436 = 718.
    2. 14 students do both, so the events can happen together: they are not mutually exclusive.
    718
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