Sample Spaces
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- Sample Spaces - Video - Sample Space, Sample Points, Events Watch
Theory
Probability is the part of maths that studies the chance of events happening. The sample space of a chance experiment is the list of every possible outcome, each written once. This Year 7 maths page shows how to write a sample space, list the outcomes in an event, and decide whether the outcomes are equally likely.
A chance experiment is an activity with an uncertain result, such as tossing a coin, rolling a die or spinning a spinner. One go is a trial, and each possible result is an outcome.
The sample space, \(S\), is the set of all the possible outcomes. Write it in curly brackets. A spinner with \(5\) sectors numbered \(1\) to \(5\) has \(S = \{1, 2, 3, 4, 5\}\): \(5\) outcomes.
An event is one or more outcomes from the sample space. On a die, the event “a number less than \(3\)” is \(\{1, 2\}\). These are the favourable outcomes for that event. An event can have no outcomes at all: on a die, “a number greater than \(6\)” is empty.
Outcomes are equally likely when each one has the same chance: a fair coin or die, equal sectors with a different label on each, or the same number of each object in a bag.
A label that appears more than once is still one outcome. List it once. (It does make that outcome more likely.)
A standard deck has \(52\) playing cards in \(4\) suits of \(13\): hearts and diamonds are red, clubs and spades are black. Each suit has an ace, the numbers \(2\) to \(10\), a jack, a queen and a king. The jack, queen and king are the picture cards.
For a two-step experiment (such as a coin and a spinner), list every pair in an orderly way, for example in a table: first result down the side, second result across the top.
Some sample spaces:
| Experiment | Sample space | Outcomes |
|---|---|---|
| roll a \(4\)-sided die | \(\{1, 2, 3, 4\}\) | \(4\) |
| spin the R, Y, W spinner | \(\{\text{R}, \text{Y}, \text{W}\}\) | \(3\) |
| choose a season | {summer, autumn, winter, spring} | \(4\) |
| choose a day of the week | {Mon, Tue, ..., Sun} | \(7\) |
| first letter of a person’s name | {a, b, c, ..., z} | \(26\) |
How to write a sample space
- Read what is recorded (a number, a colour, a letter, a pair).
- List every possible result once, in order, inside curly brackets.
- Count the outcomes.
- Decide if they are equally likely: equal sectors? the same number of each object?
How to list the outcomes in an event
- Start from the sample space.
- Test each outcome against the event (every condition must be true).
- List the ones that pass, or count them.
(i) The first letter of a person’s name.
(ii) Spinning the \(7\)-colour spinner in Diagrams.
(i) \(S = \{\text{a}, \text{b}, \ldots, \text{z}\}\): \(26\) outcomes. Not equally likely: far fewer names start with x or q than with s.
(ii) \(7\) outcomes. Not equally likely: some sectors are much bigger than others.
(i) Write the sample space.
(ii) How many outcomes are there?
(iii) Are the outcomes equally likely?
(i) \(S =\) {red, blue, yellow, green, purple}
(ii) \(5\)
(iii) Yes: \(5\) equal sectors with one colour each.
(i) List the outcomes in the event “a multiple of \(4\)”.
(ii) How many outcomes are in the event “a factor of \(12\)”?
(iii) How many outcomes are in the event “an odd number greater than \(6\)”?
\(S = \{1, 2, 3, \ldots, 12\}\)
(i) Count by \(4\)s: \(\{4, 8, 12\}\)
(ii) \(1, 2, 3, 4, 6, 12\): \(6\) outcomes
(iii) Both must be true: \(7, 9, 11\): \(3\) outcomes
(i) What is the sample space?
(ii) How many outcomes are there?
(iii) Are the outcomes equally likely?
(i) A appears three times, but it is one outcome: \(S = \{\text{C}, \text{A}, \text{R}, \text{V}, \text{N}\}\)
(ii) \(5\)
(iii) No: there are \(3\) A cards but only \(1\) C card.
(i) Are the colours equally likely?
(ii) Which change keeps them equally likely? A: add \(1\) red. B: take out \(1\) of each colour. C: add \(2\) green.
(iii) After change C, what is the sample space?
(i) Yes: \(4\) of each colour.
(ii) A gives \(5, 4, 4\); B gives \(3, 3, 3\); C gives \(4, 4, 4, 2\). Only B.
(iii) A new colour is a new outcome: {red, blue, yellow, green}
(i) List the sample space.
(ii) List the outcomes in the event “a tail and an odd number”.
Heads with each number, then tails with each number:
(i) \(\{\text{H}1, \text{H}2, \text{H}3, \text{T}1, \text{T}2, \text{T}3\}\): \(2 \times 3 = 6\) outcomes
(ii) \(\{\text{T}1, \text{T}3\}\)
(i) How many outcomes are in the event “a multiple of both \(3\) and \(4\)”?
(ii) List the outcomes in “the digits add to \(6\)”.
(iii) How many outcomes are in “a number greater than \(40\)”?
(i) A multiple of both \(3\) and \(4\) is a multiple of \(12\): \(12, 24, 36\). That is \(3\) outcomes.
(ii) \(\{6, 15, 24, 33\}\)
(iii) \(0\): no card is greater than \(40\).
(i) How many cards show a \(7\)?
(ii) How many black picture cards are there?
(i) One \(7\) in each of the \(4\) suits: \(4\)
(ii) \(2\) black suits, each with \(3\) picture cards: \(2 \times 3 = 6\)
Common pitfalls
Frequently asked questions
What is a sample space?
The list of all possible outcomes of an experiment, each written once, such as the numbers \(1\) to \(6\) for a die.
What is an event?
One or more outcomes from the sample space, such as an even number on a die.
What does equally likely mean?
Every outcome has the same chance, as on a fair die or a spinner with equal sectors labelled differently.
Do repeated labels count as different outcomes?
No. List each label once, but it is more likely than a label that appears once.
How many cards are in a standard deck?
\(52\): four suits of \(13\).