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Statistics & Probability · Foundation & Higher

Combined events & sample space

When two events happen together, you list all possible outcomes in a sample space diagram. You use this to find probabilities of combined events.

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Key facts to remember

  • 1A sample space lists all possible outcomes of an experiment.
  • 2For two dice or two spinners, use a grid (sample space diagram) to list all outcomes.
  • 3P(event) = number of favourable outcomes ÷ total number of outcomes.
  • 4Outcomes in a sample space are all equally likely (assuming fair dice, coins etc.).
  • 5P(A and B) = P(A) × P(B) only if A and B are independent events.
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Worked examples

Example 1

Two fair dice are rolled. Find the probability that the sum equals 7.

Working

  1. Sample space: 6 × 6 = 36 equally likely outcomes
  2. Outcomes summing to 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6 outcomes
  3. P(sum = 7) = 6/36 = 1/6
Answer1/6
Example 2

A coin is flipped and a fair 4-sided die is rolled. List the sample space and find P(H and even number).

Working

  1. Sample space: H1, H2, H3, H4, T1, T2, T3, T4 — 8 outcomes
  2. Favourable: H2, H4 — 2 outcomes
  3. P(H and even) = 2/8 = 1/4
Answer1/4
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Common mistakes

✗Listing some outcomes twice or missing some — use a systematic grid.
✗Not recognising that all outcomes in a fair sample space are equally likely.
✗Confusing "and" (both events occur) with "or" (at least one occurs).
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Exam tips

✓Draw a sample space grid for two-event problems — it prevents missing outcomes.
✓Count outcomes carefully: mark the favourable ones before dividing.

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