Sample space diagrams KS3 maths use a grid to list every possible outcome when two events happen together, such as rolling two dice or flipping a coin and rolling a die. Once every outcome is listed, you can count how many satisfy a condition and divide by the total to find a probability.
What is a sample space diagram?
A sample space diagram is a table or grid that lists every possible outcome of two combined events. One event's outcomes run along the top of the grid, the other event's outcomes run down the side, and each cell inside the grid shows the combined result of pairing that row and column.
The full set of every possible outcome is called the sample space. Writing it out as a grid, rather than a long list, makes it much easier to count outcomes without missing any or repeating any by accident.
How do you draw a sample space diagram — step by step?
Follow these steps whenever you need to build a sample space diagram from scratch:
- Identify the two events being combined, and list every possible outcome for each one separately.
- Draw a grid with the outcomes of the first event as column headings across the top, and the outcomes of the second event as row headings down the side.
- Fill in each cell with the combined outcome for that row and column — for two dice, this is usually written as an ordered pair such as (3, 5), or as the sum of the two values if the question asks for totals.
- Count the total number of cells in the grid — this equals the total number of equally likely outcomes in the sample space.
- Circle or shade the cells that match the event you are interested in, then count them to answer the probability question.
Worked example: two dice sample space diagram
Two fair six-sided dice are rolled together, and their scores are added. Draw a sample space diagram to find the probability that the total is 8.
Step 1 — list outcomes for each die. Each die can show 1, 2, 3, 4, 5, or 6.
Step 2 — build the grid, with Die A along the top and Die B down the side, filling each cell with the total (Die A + Die B):
| + | 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 |
Step 3 — count the total number of outcomes. The grid has 6 × 6 = 36 cells, so there are 36 equally likely outcomes.
Step 4 — count the outcomes that give a total of 8. Looking through the grid, the total 8 appears 5 times: (2,6), (3,5), (4,4), (5,3), (6,2).
Step 5 — calculate the probability. $$P(\text{total} = 8) = \frac{5}{36}$$
How do sample space diagrams handle non-numeric outcomes?
Not every combined event involves numbers. A sample space diagram works equally well for combined events like tossing a coin and rolling a die, where the outcomes are a mixture of letters and numbers.
Worked example: A fair coin is flipped and a fair six-sided die is rolled at the same time. Draw a sample space diagram and find the probability of getting a Head and an even number.
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| H | H1 | H2 | H3 | H4 | H5 | H6 |
| T | T1 | T2 | T3 | T4 | T5 | T6 |
There are 2 × 6 = 12 equally likely outcomes in total. The outcomes with a Head and an even number are H2, H4, and H6 — 3 outcomes. So:
$$P(\text{Head and even}) = \frac{3}{12} = \frac{1}{4}$$
How do you use a sample space diagram to find a probability?
Once a sample space diagram is complete, finding a probability is always the same two-part process:
- Count the number of favourable outcomes — the cells in the grid that match the event described in the question.
- Count the total number of outcomes — the total number of cells in the grid.
Then divide the favourable count by the total count to get the probability, exactly as with any probability calculation:
$$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$$
Why use a grid instead of listing outcomes as a list?
A written list of every combined outcome can easily miss a pairing or repeat one by mistake, especially once the number of outcomes climbs into the dozens. A grid organises outcomes systematically by row and column, so every combination is generated exactly once and none are skipped — this is the main reason sample space diagrams are the preferred method for combined events at KS3, ahead of tree diagrams, whenever both events have a manageable, listable number of outcomes.
| Method | Best used when | Main advantage |
|---|---|---|
| Sample space diagram (grid) | Two events, each with a small number of outcomes | Every combination shown clearly at once; easy to count |
| Tree diagram | Two or more successive events, especially with different probabilities | Shows the sequence of events and works well with unequal probabilities |
| Written list | Very few outcomes only | Quick for tiny sample spaces, but error-prone for larger ones |
Frequently asked questions
What is the difference between a sample space and a sample space diagram?
The sample space is the complete set of all possible outcomes of an experiment, whether combined or single. A sample space diagram is one way of displaying that sample space — specifically a grid layout used for combined events — so listing outcomes in a simple list is also technically a sample space, just not shown as a diagram.
Do sample space diagrams only work with dice?
No — sample space diagrams work for any combination of two events with a countable number of outcomes, including two coins, a coin and a spinner, or two spinners with different numbers of sections. Dice are simply the most common example used in KS3 maths because the outcomes are easy to list and the grid stays a manageable size.
How do you find the total number of outcomes without drawing the whole grid?
Multiply the number of possible outcomes for the first event by the number of possible outcomes for the second event — for two dice this is 6 × 6 = 36, and for a coin and a die it is 2 × 6 = 12. This shortcut works because a sample space diagram is essentially a multiplication grid of every pairing.
Can a sample space diagram be used for three combined events?
A standard grid-style sample space diagram only works cleanly for two combined events, since it needs one axis for each. For three or more combined events, a tree diagram or a systematic list is usually clearer, though some three-event problems can still be handled by treating two events as a single combined event first and pairing that with the third.
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