Sample Space Diagrams –

Sample space diagrams are a key probability tool tested on AQA, Edexcel and OCR GCSE papers at Foundation and Higher tier. They let you list every possible outcome when two events are combined — for example, rolling two dice or spinning two spinners — and then calculate probabilities by counting favourable outcomes. Getting the diagram right guarantees you will not miss any outcomes, which is the most common source of errors. This guide walks you through the method, provides worked examples and highlights the pitfalls examiners look for. For a full overview of the specification, see our complete GCSE Maths topics list.

What Is a Sample Space Diagram?

A sample space diagram is a systematic way of listing all possible outcomes for two combined events. The most common form is a two-way grid (or table) where one event runs along the top and the other down the side. Each cell in the grid shows the combined outcome.

Key Formulas

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

Step-by-Step Method

1. List the outcomes of each event along the top and down the side of a grid.
2. Fill in every cell with the combined result (e.g. the sum, product or pair).
3. Count the total number of outcomes (this should match the product rule).
4. Count the favourable outcomes — those that satisfy the condition in the question.
5. Calculate the probability as favourable ÷ total and simplify.

Worked Example 1 — Foundation Level

Question: Two fair six-sided dice are rolled and their scores are added together. Find the probability that the total is 7.

Working:

Draw a 6 × 6 grid. The sums are:

Total outcomes = 36. Outcomes that give a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6.

Answer: P(total is 7) = 6/36 = 1/6.

Worked Example 2 — Higher Level

Question: A fair 4-sided spinner (1, 2, 3, 4) and a fair 6-sided dice are used. The scores are multiplied. Find the probability that the product is greater than 12.

Working:

Draw a 4 × 6 grid of products:

Total outcomes = 4 × 6 = 24.

Products greater than 12: 15, 18, 16, 20, 24 = 5 outcomes.

Answer: P(product > 12) = 5/24.

Worked Example 3 — Exam Style

Question: Two fair coins and a fair six-sided dice are used. A coin is flipped and a dice is rolled. The sample space for the coin shows H and T; the dice shows 1–6. Find the probability of getting a head and an even number.

Working:

Total outcomes = 2 × 6 = 12.

Favourable outcomes: (H,2), (H,4), (H,6) = 3 outcomes.

Answer: P(head and even) = 3/12 = 1/4.

Common Mistakes

• Missing outcomes. If you do not draw a complete grid, you risk leaving out combinations. Always check that the number of cells matches the product rule.
• Double-counting. When listing pairs like (2,5) and (5,2), these are different outcomes for two dice — count them both.
• Confusing sums and products. Read the question carefully to see whether you should add, multiply or simply list the pair.

Exam Tips

• A well-drawn grid is quick and almost guarantees full marks — spend the time setting it up neatly.
• For two dice, the total is always 36 outcomes. Memorise the symmetry: the most likely sum is 7 (six ways), with sums of 2 and 12 each having only one way.
• If asked for P(at least 9), count outcomes that give 9, 10, 11 and 12.
• For extensions using tree diagrams, see probability tree diagrams.

Practice Questions

Q1 (Foundation): Two fair six-sided dice are rolled and the scores added. Find the probability that the total is less than 5.

Q2 (Foundation): A fair coin is flipped and a fair four-sided dice (1–4) is rolled. List the sample space and find P(tail and number > 2).

Q3 (Higher): Two fair six-sided dice are rolled and the scores are multiplied. Find the probability that the product is a square number.

Practise sample space diagrams and more for free on GCSEMathsAI.

Related Topics

• Probability Basics and Relative Frequency
• Probability Tree Diagrams
• Combined Events Probability

Summary

A sample space diagram is a two-way grid that lists every possible outcome for two combined events. Fill in each cell with the sum, product or pair depending on the question. The total number of outcomes equals the product of the individual outcomes (e.g. 6 × 6 = 36 for two dice). To find a probability, count the favourable outcomes and divide by the total. Always draw the full grid to avoid missing outcomes, and simplify your fraction at the end.