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Sample space diagrams

List combined outcomes systematically in a sample space diagram and use the grid to calculate probabilities.

Probability and StatisticsFoundation and HigherGrades 4 to 6Focused skill

Curriculum path: GCSE Maths > Edexcel > Probability and Statistics > Sample space diagrams

Pearson Edexcel GCSE Maths Probability P6 and P7: enumerate outcomes systematically using tables and construct theoretical possibility spaces.

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Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

A sample space diagram shows all possible outcomes for two events, such as rolling two dice or choosing from two spinners.

Use rows for one event and columns for the other event. Fill every cell so no outcome is missed.

If every cell is equally likely, probability = number of successful cells / total number of cells.

For two fair six-sided dice there are 36 equally likely outcomes because 6 x 6 = 36.

In exam questions, the event may be written in words, such as total is greater than 8 or both numbers are the same. Mark the successful cells carefully.

A sample space diagram is different from a probability tree. Use a grid when the question asks for combinations from two lists of outcomes.

Key ruleFor equally likely grid outcomes, P(event) = successful cells / total cells.

Diagram guide

112233445566234567345678456789567891067891011789101112highlighted cells add to 5
Sample space gridRows and columns create every combined outcome; the highlighted cells match the event.

Worked examples

Use a dice sample space

Two fair dice are rolled. Find the probability that the total is 5.

112233445566234567345678456789567891067891011789101112highlighted cells add to 5
Two dice totalsFind the cells where the two dice add to 5.
  1. There are 6 x 6 = 36 equally likely outcomes.
  2. Totals of 5 are: 1 + 4, 2 + 3, 3 + 2 and 4 + 1.
  3. There are 4 successful outcomes.
  4. P(total is 5) = 4 / 36 = 1 / 9.

Answer: 1/9

Both outcomes the same

Two fair dice are rolled. Find the probability of a double.

112233445566234567345678456789567891067891011789101112highlighted cells add to 5
Sample space gridRows and columns create every combined outcome; the highlighted cells match the event.
  1. A double means both dice show the same number.
  2. The successful outcomes are (1,1), (2,2), (3,3), (4,4), (5,5), (6,6).
  3. There are 6 successful outcomes out of 36.
  4. 6 / 36 = 1 / 6.

Answer: 1/6

Greater than a number

Two fair dice are rolled. Find the probability that the total is greater than 10.

112233445566234567345678456789567891067891011789101112highlighted cells add to 5
Sample space gridRows and columns create every combined outcome; the highlighted cells match the event.
  1. The totals greater than 10 are 11 and 12.
  2. Total 11 happens with (5,6) and (6,5).
  3. Total 12 happens with (6,6).
  4. There are 3 successful outcomes out of 36.

Answer: 3/36 = 1/12

Common mistakes

  • Counting only one order, such as 1 + 4 but not 4 + 1.
  • Using 12 as the denominator for two dice instead of 36.
  • Missing outcomes because the grid is not completed systematically.
  • Simplifying the fraction incorrectly.
  • Using a sample space grid when outcomes are not equally likely without adjusting for probability.

Quick exercise

Try these before moving to the exam-style questions.

  1. Two fair coins are flipped. How many outcomes are in the sample space?
    112233445566234567345678456789567891067891011789101112highlighted cells add to 5
    Sample space gridRows and columns create every combined outcome; the highlighted cells match the event.
  2. Two fair dice are rolled. How many outcomes are in the sample space?
    112233445566234567345678456789567891067891011789101112highlighted cells add to 5
    Sample space gridRows and columns create every combined outcome; the highlighted cells match the event.
  3. Two fair dice are rolled. How many outcomes give a total of 2?
    112233445566234567345678456789567891067891011789101112highlighted cells add to 5
    Sample space gridRows and columns create every combined outcome; the highlighted cells match the event.
  4. Two fair dice are rolled. How many outcomes give a total of 7?
    112233445566234567345678456789567891067891011789101112highlighted cells add to 5
    Sample space gridRows and columns create every combined outcome; the highlighted cells match the event.
  5. Two fair dice are rolled. Find P(double).
    112233445566234567345678456789567891067891011789101112highlighted cells add to 5
    Sample space gridRows and columns create every combined outcome; the highlighted cells match the event.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleFoundation and HigherNon-calculator3 marks

Two fair dice are rolled. Work out the probability that the total is 4.

112233445566234567345678456789567891067891011789101112highlighted cells add to 5
Question gridUse the completed grid to count totals of 4.
sample spacedice probabilityfoundation probability
Standard exam styleFoundation and HigherNon-calculator4 marks

Spinner A has numbers 1, 2, 3 and Spinner B has numbers 2, 4, 6. Each spinner is fair. Work out the probability that the product of the two numbers is greater than 10.

112233445566234567345678456789567891067891011789101112highlighted cells add to 5
Sample space gridRows and columns create every combined outcome; the highlighted cells match the event.
sample spacespinnercombined outcomes
ChallengeHigherEither5 marks

Two fair dice are rolled. Event A is 'the total is at least 10'. Event B is 'the two numbers are the same'. Find P(A or B).

112233445566234567345678456789567891067891011789101112highlighted cells add to 5
Sample space gridRows and columns create every combined outcome; the highlighted cells match the event.
sample spaceor probabilityoverlap