Sample Space & Outcomes
Learn about the sample space and outcomes for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-07). The sample space lists every outcome a chance experiment can produce, such as rolling a die, tossing a coin or spinning a spinner.
You will write a sample space in curly brackets with each outcome listed once, count how many outcomes an experiment has, and count the outcomes that fit an event such as “even” — the groundwork for every probability you will calculate in games, raffles and everyday chance.
Theory
The sample space \(S\) is the list of every possible outcome of a chance experiment. This Year 12 Standard 1 (NSW) guide shows how to write a sample space in curly brackets, count the outcomes \(n(S)\), and count the outcomes that fit an event such as “even” or “greater than \(4\)”.
A chance experiment is something like rolling a die, tossing a coin, spinning a spinner or drawing a card from a box. An outcome is one possible result of the experiment.
The sample space \(S\) is the list of every possible outcome, written once each inside curly brackets. For a coin, \(S=\{H,\,T\}\). The number of outcomes in the sample space is written \(n(S)\): a die has \(S=\{1,2,3,4,5,6\}\), so \(n(S)=6\).
To count the outcomes that fit a description (even, greater than \(4\), a multiple of \(5\)), list them from the sample space and then count them. Outcomes are equally likely when each one has the same chance — a fair die or coin, equal sectors, identical cards.
The sample space lists every outcome once, and \(n(S)\) counts them:
An event is part of the sample space. List its outcomes, then count them:
How to write a sample space and count outcomes
- List every possible result of the experiment inside curly brackets, e.g. \(S=\{A,B,C,D,E\}\).
- Check that nothing is missing, nothing impossible is included, and each outcome is written only once.
- Count the outcomes: \(n(S)\) for the whole sample space, or list and count just the outcomes that fit the description.
Each sector is one outcome; list each label once.
| \(S\) | \(=\) | \(\{A,B,C,D,E\}\) |
| \(n(S)\) | \(=\) | \(5\) |
\(S=\{A,B,C,D,E\}\) with \(5\) outcomes.
List the multiples of \(3\) inside the sample space, then count them.
| \(n(S)\) | \(=\) | \(15\) |
| \(\text{multiples of }3\) | \(=\) | \(\{3,6,9,12,15\}\) |
| \(n(\text{multiples of }3)\) | \(=\) | \(5\) |
\(5\) of the outcomes are multiples of \(3\).
There are \(6\) cards, but the letter \(Y\) appears twice. List each letter only once.
| \(\text{cards}\) | \(:\) | \(S,\,Y,\,D,\,N,\,E,\,Y\) |
| \(\text{sample space}\) | \(=\) | \(\{S,Y,D,N,E\}\) |
| \(\text{number of outcomes}\) | \(=\) | \(5\) |
The sample space is \(\{S,Y,D,N,E\}\) with \(5\) outcomes. (\(Y\) is more likely than the other letters, because it is on two cards.)
Write the sample space once, then use it for every part.
| \(\text{(i) } S\) | \(=\) | \(\{1,2,3,\dots,12\}\) |
| \(n(S)\) | \(=\) | \(12\) |
| \(\text{(ii) odd and } > 6\) | \(=\) | \(\{7,9,11\}\) |
| \(n\) | \(=\) | \(3\) |
(iii) The discs are identical, so each number has the same chance.
(i) \(12\) (ii) \(3\) (iii) Yes, they are equally likely.
Common pitfalls
Frequently asked questions
What is a sample space in probability?
The sample space is the list of every possible outcome of a chance experiment, written once each inside curly brackets. For one roll of a die it is S = {1, 2, 3, 4, 5, 6}.
What is an outcome in probability?
An outcome is one possible result of a chance experiment. Getting a 4 when you roll a die, or heads when you toss a coin, are both outcomes.
What does n(S) mean?
n(S) is the number of outcomes in the sample space. A die has 6 outcomes, so n(S) = 6. A coin has 2 outcomes, so n(S) = 2.
How do you write a sample space?
List every possible result inside curly brackets, separated by commas, and write each one only once. For a spinner with sectors A to E the sample space is S = {A, B, C, D, E}.
Are all outcomes in a sample space equally likely?
Not always. They are equally likely only when each has the same chance, such as a fair die or equal sectors. A spinner with three green sectors and one white sector does not give equally likely colours.
What is the sample space for tossing a coin?
Tossing one coin has two outcomes, heads and tails, so the sample space is S = {H, T} and n(S) = 2.