Sample space diagrams and listing outcomes

When two things happen together, a grid or a systematic list shows every equally likely outcome, so no case is missed.

Practise this lesson free (6 questions)

Listing systematically

Flip a coin and roll a dice: H1, H2, ..., H6, T1, ..., T6 gives 2 × 6 = 12 outcomes.
Three coins: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT gives 2 × 2 × 2 = 8 outcomes.
Fix the first choice, run through the rest, then change the first.

Sample space grid

Roll two dice and add the scores.
+123456
1234567
2345678
3456789
45678910
567891011
6789101112

Probabilities from the grid

There are 36 equally likely outcomes.
P(total 7) = 636 = 16
P(total 10 or more) = 636 = 16
Count the cells that fit, and divide by the total number of cells.
Worked example

Two fair dice are rolled. Find P(total is 7).

  1. Count the 7s in the grid: 6
  2. P = 636 = 16

Answer: 16

Key idea

List outcomes in a fixed order or use a grid. Every cell is equally likely, so P = cells that fit ÷ total cells.

Extra practice

2 more questions

Try each one, then tap to check.

  1. Two fair dice are rolled and the scores added. Find P(total is 9 or more). Give your answer as a fraction in its simplest form.

    Show the answer

    Answer: 5/18

    10 of the 36 cells in the sample space: 1036 = 518.

  2. Two fair dice are rolled and the scores added. Find P(total is 9). Give your answer as a fraction in its simplest form.

    Show the answer

    Answer: 1/9

    4 of the 36 cells in the sample space: 436 = 19.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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