Sample Space Diagrams
A sample space diagram is a table, often a grid, that lists every possible outcome when two events happen together, such as rolling two dice or spinning two spinners, making it easy to count outcomes and calculate probabilities. Sample space diagrams appear in the Probability strand of GCSE Statistics, usually for finding the probability of combined events.
Before you start
Make sure you're comfortable with these topics first:
Method
- List all possible outcomes of the first event along one side of a grid, and all possible outcomes of the second event along the other side.
- Fill in each cell of the grid with the combined outcome (for example, a sum or product) for that row and column.
- Count the total number of cells in the grid; this is the total number of equally likely outcomes.
- Identify and count the cells that satisfy the event you are interested in.
- Write the probability as (number of successful cells) divided by (total number of cells), simplifying the fraction.
- Use the completed diagram to answer follow-up questions without having to relist the outcomes each time.
Worked example
Two fair four-sided dice, each numbered 1 to 4, are rolled and their scores added together. Draw a sample space diagram of the totals and find the probability that the total is 5.
- Draw a 4 by 4 grid with dice A's scores (1, 2, 3, 4) along the top and dice B's scores (1, 2, 3, 4) down the side.
- Fill each cell with the sum, so row 2 column 3 gives 2 + 3 = 5.
- The grid of totals runs from 2 (1+1) up to 8 (4+4).
- Total number of outcomes = 4 x 4 = 16.
- Cells showing a total of 5 are (1,4), (2,3), (3,2) and (4,1), which is 4 cells.
- Final answer: P(total is 5) = 4/16 = 1/4.
Practice questions
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Q1Two fair coins are flipped. How many equally likely outcomes are in the sample space?Show answer
Answer: 4 (HH, HT, TH, TT)
Q2Using the two-coin sample space, find P(exactly one head).Show answer
Answer: 1/2 (2 out of 4: HT, TH)
Q3Two fair six-sided dice are rolled and the scores are multiplied together. How many cells are in the sample space diagram?Show answer
Answer: 36 (6 x 6)
Q4Using two fair six-sided dice added together, how many ways can you score a total of 7?Show answer
Answer: 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1)
Q5Using two fair six-sided dice added together, find P(total is 7).Show answer
Answer: 1/6 (6/36)
Q6A spinner with sections A, B and C is spun alongside a fair coin. How many outcomes are in the sample space, and what is P(getting C and Heads)?Show answer
Answer: 6 outcomes; P(C and Heads) = 1/6
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
Two fair four-sided dice, numbered 1 to 4, are rolled and their scores are added together. Using a sample space diagram, work out the probability that the total is 8.
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A fair coin is flipped and a fair six-sided dice is rolled at the same time. (a) Draw a sample space diagram to show all the possible outcomes. (b) Find the probability of getting a Head and an even number.
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Two fair five-sided spinners, each numbered 1 to 5, are spun and their scores multiplied together. (a) Complete a sample space diagram to show all 25 possible products. (b) Find the probability that the product is greater than 15.
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the sample space diagrams worksheet pack - 18 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 18 of GCSE Statistics Foundation Workbook 2 and chapter 24 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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