Probability becomes unreliable when outcomes are listed casually. A sample space diagram is a control surface for making every possible result visible exactly once.
This guide develops sample spaces as a systematic representation. It shows how tables and organised lists prevent missing outcomes, how to distinguish equally likely from unequally likely results, and how sample spaces connect directly to probability, tree diagrams and counting.
One-sentence answer
A sample space is the complete set of possible outcomes for a probability experiment, organised so that counting and comparison become reliable.
Why organisation matters
If two coins are tossed, writing HH, HT, TH, TT is safe because the four ordered outcomes are visible. Writing “two heads, one head, no heads” describes categories, not equally likely basic outcomes. The category “one head” contains two outcomes.
Probability depends on the correct level of representation.
Worked example 1: two coins
HH HT
TH TTFor fair independent coins, all four outcomes are equally likely. Therefore P(exactly one head) = 2/4 = 1/2.
Worked example 2: two dice
Rolling two fair six-sided dice gives 36 ordered pairs. A 6×6 table is more reliable than trying to remember every pair.
To find the probability of total 7, count:
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)So P(total 7) = 6/36 = 1/6.
This also explains why sums are not equally likely. There is only one way to make total 2, but six ways to make total 7.
Worked example 3: spinner and coin
A fair spinner has three equal sectors A, B and C. A fair coin is tossed. The sample space is:
AH, AT, BH, BT, CH, CTThere are six equally likely combined outcomes. P(B or C with heads) means BH or CH, so the probability is 2/6 = 1/3.
Ordered outcomes versus unordered descriptions
In repeated experiments, order often matters. Red then blue is a different path from blue then red even if the final colour counts are the same.
When the experiment itself distinguishes first and second positions, the sample space should preserve that order.
Unequally likely outcomes
A sample space can list all possible outcomes without making them equally likely. Suppose a spinner has three sectors labelled A, B and C, but A covers half the circle while B and C each cover a quarter. The sample space is still {A, B, C}, but their probabilities are 1/2, 1/4 and 1/4.
Do not divide by the number of labels unless the model justifies equal likelihood.
Systematic tables
Tables are especially useful when two variables combine. Put the outcomes of the first event along the rows and the second event along the columns. Each cell then represents one combined result.
This makes completeness visible: if every row and column has been filled, missing combinations become less likely.
Restrictions change the sample space
If a question says “choose two different digits from 1, 2, 3”, then outcomes such as (1,1) are not permitted. The sample space must reflect the rule of the experiment, not a generic table copied from an earlier problem.
From sample space to event
An event is a subset of the sample space. If two dice are rolled and event A is “sum greater than 9”, then A contains all ordered pairs giving 10, 11 or 12.
This language makes probability precise: the experiment defines the sample space; the question defines the event inside it.
Diagnostic table
| Observed mistake | Likely issue | Repair |
|---|---|---|
| Misses combinations | Unsystematic listing | Use a table or fixed-order list |
| Counts HT and TH as one outcome | Order confusion | Preserve sequence positions |
| Assumes all labels equally likely | Model confusion | Check sector sizes or stated probabilities |
| Includes forbidden repeated choices | Constraint ignored | Build the rule into the sample space |
| Uses sums as basic equally likely outcomes | Representation too compressed | Return to ordered pairs |
Practice
- List the sample space for a fair coin and a fair spinner labelled 1, 2.
- Two fair dice are rolled. Find P(total 4).
- Two fair coins are tossed. Find P(at least one head).
- Digits 1, 2, 3 are used to form a two-digit number without repetition. List the sample space.
Answers
1. H1, H2, T1, T2. 2. 3/36 = 1/12, from (1,3), (2,2), (3,1). 3. 3/4. 4. 12, 13, 21, 23, 31, 32.
Connected routes
Start with Probability as a Model of Uncertainty, then continue to Tree Diagrams Without Double-Counting for sequential events. Return to the Mathematics Learning Hub for the wider Secondary Mathematics route.