Sample space diagrams
A sample space diagram is a table that shows every possible outcome when two things happen together, such as rolling two dice. One event goes across the top, the other down the side, and each cell is one outcome. If the outcomes are equally likely, the probability of an event is the number of cells that match it divided by the total number of cells.
What is a sample space?
The sample space is the list of all possible outcomes of an experiment. For one coin it is just heads and tails. For two coins it is HH, HT, TH and TT: four outcomes, not three, because getting a head on the first coin and a tail on the second is different from the other way round. A sample space diagram is a tidy way to list combined outcomes so none are missed or counted twice.
AQA's GCSE Maths specification asks students to list outcomes systematically using tables and grids, and to construct possibility spaces for combined experiments with equally likely outcomes, then use them to calculate probabilities. "Possibility space" means the same thing as sample space.
How to draw a sample space diagram
- Write the outcomes of the first event along the top row.
- Write the outcomes of the second event down the first column.
- Fill each cell with the combined outcome, or with the result you care about: the total, the product, the difference.
- Count the cells. The total number of cells is the number of equally likely outcomes.
Example 1: the total of two dice
Two fair six-sided dice are rolled and the scores added. The first dice goes across the top and the second down the side:
| + | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
There are 6 × 6 = 36 equally likely outcomes. Now count the cells for each question:
- P(total of 7): the 7s run along a diagonal, 6 cells. 6/36 = 1/6.
- P(total of 8): 5 cells, so 5/36.
- P(total of 10 or more): three 10s, two 11s and one 12, 6 cells. 6/36 = 1/6.
- P(a double): the cells where both dice match run down the other diagonal, 6 cells. 1/6.
The table also shows why 7 is the most likely total: there are more ways to make it than any other number. The totals from 2 to 12 are not equally likely, so you can't say P(7) = 1/11.
Try finding a sample space yourself, then press Mark to check it in the course.
Type the number of outcomes and select those matching an event.
Press Mark to have your answer marked in the Probability course.
Example 2: a coin and a spinner
A fair coin is flipped and a fair spinner numbered 1 to 4 is spun. The cells show the combined outcome:
| 1 | 2 | 3 | 4 | |
|---|---|---|---|---|
| Head | H1 | H2 | H3 | H4 |
| Tail | T1 | T2 | T3 | T4 |
There are 2 × 4 = 8 outcomes. A head with an even number is H2 or H4, so P(head and even) = 2/8 = 1/4. A tail or a 1 is T1, T2, T3, T4 or H1, so P(tail or 1) = 5/8: count H1 only once even though it fits the "1" part.
Example 3: the difference between two dice
This time each cell holds the bigger score minus the smaller one:
| − | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 0 | 1 | 2 | 3 | 4 | 5 |
| 2 | 1 | 0 | 1 | 2 | 3 | 4 |
| 3 | 2 | 1 | 0 | 1 | 2 | 3 |
| 4 | 3 | 2 | 1 | 0 | 1 | 2 |
| 5 | 4 | 3 | 2 | 1 | 0 | 1 |
| 6 | 5 | 4 | 3 | 2 | 1 | 0 |
A difference of 2 appears in 8 cells, so P(difference of 2) = 8/36 = 2/9. A difference of 0 is a double, 6/36 = 1/6, matching example 1.
Expected frequency
Once you have a probability, multiply by the number of trials to find how many times you'd expect it. If two dice are rolled 180 times, a total of 7 is expected 180 × 1/6 = 30 times. Real results will vary a little either side. AQA's specification asks students to use equally likely outcomes to calculate expected outcomes of repeated experiments like this.
Sample space diagram, tree diagram or list?
| Method | Best for |
|---|---|
| List | Small sample spaces, such as two coins |
| Sample space diagram | Two events with several equally likely outcomes, such as two dice |
| Tree diagram | Events with different probabilities, or three or more stages |
| Venn diagram | Overlapping groups, such as students who study French or Spanish |
The Probability course covers all of these, and the probability calculator checks combined probabilities.
Common mistakes
- Treating (3, 4) and (4, 3) as the same outcome. They are different cells, which is why there are 36 outcomes, not 21.
- Assuming every total is equally likely. Only the cells are equally likely; the totals are not.
- Counting a cell twice in an "or" question, such as H1 in example 2.
- Using the wrong total. The denominator is the number of cells, such as 36 for two dice or 8 for a coin and a four-sided spinner.
To practise probability under time pressure, try today's GCSE Maths Foundation or GCSE Maths Higher short timed paper. Each paper mixes topics.
Common questions
What is a sample space diagram?
How many outcomes are there with two dice?
What is the most likely total with two dice?
What is the probability of rolling a double?
Is a sample space diagram the same as a possibility space?
When should I use a tree diagram instead?
Revision cards
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