P2ProbabilityFoundation

Sample Spaces & Listing

A sample space is a list of all the possible outcomes. Listing them carefully — in an organised way — makes sure you do not miss any, which is the secret to combined-event probability questions.

30 min AQA GCSE Maths

What you'll learn

List outcomes systematically
Draw a sample space diagram
Count outcomes for two events
Find probabilities from a sample space

Systematic listing

When listing outcomes, work in an order (smallest first, or fix one item and vary the other) so you never miss one or repeat one. For two dice, a $sample space diagram$ (a grid) shows all 36 outcomes neatly.

1Each dice has 6 outcomes.

6 \times 6 = 36

Answer36 outcomes

Tip — For two events, a grid (sample space diagram) is faster and safer than a long list.

Finding probabilities

Once every outcome is listed, count the ones you want and divide by the total. For two dice, P(total of 7) is the number of ways to make 7 divided by 36.

Remember these

outcomes = m \times n

m outcomes then n outcomes.

P = \frac{wanted outcomes}{total in sample space}

Probability from a list.

Watch out for these

Listing outcomes randomly and missing some.

List systematically (in order) so you catch every outcome.

Counting (1,2) and (2,1) as the same outcome for two dice.

They are different outcomes in the sample space.

Key takeaways

List all outcomes systematically.
Use a grid (sample space diagram) for two events.
P = wanted ÷ total number of outcomes.

Test yourself

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