All subjects · Mathematics · WST01 (S1)

Sample spaces and elementary probability

Probability measures how likely an event is, on a scale from 0 to 1. Listing the sample space is the safest way to find probabilities for equally likely outcomes.

Key words

Experiment: a repeatable process, such as throwing two dice.
Sample space: the set of all possible outcomes.
Event: a set of outcomes, such as 'the total is 7'.

For equally likely outcomes: P(A) = number of outcomes in A ÷ total number of outcomes.

Sample space diagrams

For two dice there are 6 × 6 = 36 equally likely outcomes. A grid of totals makes counting easy: a total of 7 happens in 6 ways, so P(7) = 6/36 = 1/6.

+	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

Totals when two fair dice are thrown

Complements and expected frequency

The complement A' means 'not A': P(A') = 1 - P(A). It is often quicker: P(at least one six in two throws) = 1 - P(no sixes) = 1 - 25/36 = 11/36.

Expected frequency = number of trials × probability.

Worked example

Two fair dice are thrown. Find the probability that the product is even.

The product is odd only if both dice are odd: 3 × 3 = 9 outcomes.
P(odd) = 9/36 = 1/4.
P(even) = 1 - 1/4

Answer: 3/4

Key idea: List the sample space; P(A) = favourable ÷ total for equally likely outcomes; P(A') = 1 - P(A).

Check yourself

Two fair dice are thrown. Find the probability that the total is 7. Give your answer as a fraction in its simplest form.
Show answer
1/6. 6 of the 36 outcomes total 7: 6/36 = 1/6.
Two fair dice are thrown. Find the probability that the total is 10 or more. Give your answer as a fraction in its simplest form.
Show answer
1/6. Totals 10, 11, 12 occur in 3 + 2 + 1 = 6 ways: 6/36 = 1/6.
Two fair dice are thrown. Find the probability that the product of the scores is even. Give your answer as a fraction in its simplest form.
Show answer
3/4. Odd products need two odd scores: 9 ways. So 27 ways are even: 27/36 = 3/4.
Two fair dice are thrown. Find the probability of a double (the same score on each). Give your answer as a fraction in its simplest form.
Show answer
1/6. 6 doubles out of 36: 1/6.
A fair die is thrown twice. Find the probability of at least one six. Give your answer as a fraction in its simplest form.
Show answer
11/36. P(no six) = 5/6 × 5/6 = 25/36, so P(at least one) = 11/36.
The probability that a seed germinates is 0.85. How many of 400 seeds are expected to germinate?
Show answer
340. 400 × 0.85 = 340.
P(A) = 0.37. Find P(A').
Show answer
0.63. 1 - 0.37 = 0.63.

Practise with instant feedback

Sources

Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S1

Last updated 7 November 2026. Spotted a mistake? Tell us.