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Sample Space Diagrams

FoundationHigherAQAEdexcel

Learn Sample Space Diagrams for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. Sample space diagrams list all the possible outcomes of one or two events.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA and Edexcel specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A sample space diagram lists all the possible outcomes of an event, making probabilities straightforward to count.

For two events it is drawn as a two-way grid, with the outcomes of one event across the top and the other down the side. Each cell shows the combined outcome. Two dice give a grid of 36 cells, and the total is 36 rather than 12 because each of the 6 outcomes on the first dice can pair with each of the 6 on the second.

Once the diagram is complete, the probability of any event is the number of cells satisfying it divided by the total number of cells. This turns awkward probability questions into simple counting.

Revision notes

Drawing the diagram

For two events, draw a two-way grid with one event's outcomes across the top and the other's down the side.

Each cell holds the combined outcome — for two dice this might be the sum, or the pair of values. Fill in every cell before counting.

Counting the total

The total number of outcomes is the product of the numbers of outcomes for each event.

Two dice give \(6 \times 6 = 36\) outcomes, not 12. Each outcome on the first can pair with each on the second, which is why the totals multiply.

Finding probabilities

Count the cells satisfying the event, then divide by the total number of cells.

This reduces a probability question to counting, which is far less error-prone than reasoning about combinations in the abstract.

Key points

  • A sample space diagram lists all outcomes.
  • For two events it is a two-way grid.
  • Totals multiply, so two dice give 36 outcomes.
  • Each cell shows a combined outcome.
  • Count the cells satisfying the event.
  • Divide by the total number of cells.

Worked examples

Example 1

Two fair dice are rolled. Work out the total number of possible outcomes. [2 marks]

Working

Each dice has 6 outcomes, and each outcome on the first can pair with each on the secondexplain why the totals multiply
\(6 \times 6 = 36\) outcomeswork out the total

Example 2

Two fair dice are rolled. Work out the probability that the total is 7. [3 marks]

Working

The pairs giving 7 are 1+6, 2+5, 3+4, 4+3, 5+2 and 6+1list the favourable outcomes
There are 6 favourable outcomes out of 36count them against the total
\(P(\text{total } 7) = \frac{6}{36} = \frac{1}{6}\)write and simplify the probability

Example 3

A coin is flipped and a dice rolled. Work out the total number of outcomes. [2 marks]

Working

The coin has 2 outcomes and the dice has 6state the outcomes for each event
\(2 \times 6 = 12\) outcomesmultiply to find the total

Common mistakes

  • Adding the outcomes instead of multiplying.

    Two dice give 6 × 6 = 36, not 6 + 6 = 12.

  • Missing outcomes when counting.

    Draw the full grid rather than listing from memory.

  • Treating 2+5 and 5+2 as the same outcome.

    They are different cells in the grid and both count.

  • Not simplifying the final fraction.

    Give the probability in its simplest form.

Exam tips

  • Draw the full grid rather than listing outcomes.
  • Multiply the outcome counts for the total.
  • Count ordered pairs — 2+5 and 5+2 are different.
  • Simplify the final probability.

Key terms

Sample space
The set of all possible outcomes.
Sample space diagram
A grid or list showing all outcomes.
Outcome
One possible result of an event.
Combined outcome
The result of two events happening together.

Related topics

Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.