Sample Space

Definition of Sample Space

Useful Notation

Types of Sample Spaces

Listing and Representing Outcomes

Properties of a Sample Space

Relationship to Events

Sample Space in Practice

Common Mistakes

Connections to Other Concepts

Sample Space — The First Step in Probability

As explained on the page about the probability of events, probability compares the outcomes we want to the full set of outcomes that could happen. The basic expression we already introduced there is:

Probability = (favourable outcomes) / (all possible outcomes)

Since “all possible outcomes” appears directly in the formula, it makes sense to understand what this collection actually is and how we identify it for different situations. That collection is the sample space. It describes every outcome the scenario can produce and forms the foundation for defining events and assigning probabilities.

Definition of Sample Space

The sample space is the complete collection of all possible outcomes a scenario can produce. Every outcome that can happen belongs to it, and anything that cannot happen is not included.

Each outcome represents one full, specific result of the situation. Together, these outcomes form the background on which events are defined and probabilities are assigned.

Examples:
Rolling a die → {1, 2, 3, 4, 5, 6}
Tossing two coins → {HH, HT, TH, TT}
Measuring a persons height → all real numbers in an interval (for example, [0, 3] meters)

Useful Notation

Before working with sample spaces in probability, a few standard symbols are used to describe outcomes and the sets they belong to:

(\Omega)

— the sample space (the set of all possible outcomes)

(\omega)

— a single outcome (an element of

(\Omega))

{ } — explicit listing of outcomes for finite sample spaces
Set-builder notation, e.g.

(\{x : x > 0\})

Intervals for continuous outcomes, e.g.

([0,\infty))

Cartesian products for multi-step scenarios, e.g.

(\Omega = A \times B)

Events as subsets of the sample space:

(A \subseteq \Omega)

These symbols will appear throughout the page whenever we describe outcomes or refer to parts of the sample space.

Types of Sample Spaces

Different scenarios produce different kinds of sample spaces. They generally fall into a few common categories:

Finite sample spaces: a limited number of outcomes, e.g. rolling a die
$(\Omega = {1,2,3,4,5,6})$

Countably infinite sample spaces: outcomes can be listed in sequence, e.g. number of trials until first success
$(\Omega = {1,2,3,\ldots})$

Uncountable or continuous sample spaces: outcomes fill an interval or region, e.g. measuring time or height
$(\Omega = [0,\infty))$

These categories help determine how probabilities are assigned and what tools are used to work with the sample space.

Listing and Representing Outcomes

Once the sample space is identified, it can be written in different ways depending on the scenario and the size of the outcome set:

Explicit lists for small finite spaces, e.g.
$(\Omega = {1,2,3,4,5,6})$

Ordered pairs for multi-step experiments, e.g. two coin tosses
(Omega = {(H,H), (H,T), (T,H), (T,T)})

Sequences or tuples when more than two components are involved, e.g. three dice
(Omega = {(x_1, x_2, x_3) : x_i in {1,ldots,6}})

Intervals when outcomes vary continuously, e.g. a measurement
(Omega = [0,1])

Set-builder notation for describing outcomes by a rule, e.g.
(Omega = {x : 0 le x le 10})

Cartesian products for combining simpler spaces, e.g.
(Omega = A imes B)

These representations make it easier to see how outcomes are organized and how events will be formed from them.

Properties of a Sample Space

A sample space is not just any set—it must satisfy a few basic requirements so that probability can be defined consistently:

every outcome that can occur in the scenario.
cannot occur.
mutually exclusive (only one outcome happens in a single trial).
collectively exhaustive (something from the set must occur).

These properties ensure that probabilities assigned to events make sense and behave consistently.

Relationship to Events

Events are built directly from the sample space. An event is simply a subset of (Omega) that collects the outcomes we care about in a particular question.

Viewing events as subsets of (Omega) makes probability assignments consistent and ties every event back to the underlying structure of the scenario.

Sample Space in Practice

Different situations lead to different forms of sample spaces. A few common examples show how the idea appears in everyday probability questions:

$(\Omega = {1,2,3,4,5,6})$
$(\Omega =)$ all 52 individual cards

These examples show how the structure of (Omega) changes with the scenario, but the idea remains the same: it captures every outcome the situation can produce.

Common Mistakes

A correct sample space removes ambiguity and prevents errors in later calculations.

Connections to Other Concepts

$(\Omega)$ , built directly from the sample space
$(\Omega)$ is fixed
$(\Omega)$ to numerical values
$(\Omega)$

The sample space anchors every other concept in probability, making it the natural starting point for the entire subject.