Sample Space and Events

Before assigning numbers to uncertainty, you need a precise language for describing what can happen. That language is set theory: the set of all possible outcomes, its subsets, and the operations that combine them.

Sample space

A random experiment is any process whose outcome cannot be predicted with certainty in advance. The sample space $\Omega$ is the set of all possible outcomes of the experiment.

Examples.

Experiment	Sample space $\Omega$
Flip a single coin	$\{H, T\}$
Roll a six-sided die	$\{1, 2, 3, 4, 5, 6\}$
Count packets arriving at a router in one second	$\{0, 1, 2, 3, \ldots\} = \mathbb{N}_0$
Measure the lifetime of a light bulb (hours)	$[0, \infty)$
Record the coordinates of a uniformly random point in the unit square	$[0,1]^2$

Each element $\omega \in \Omega$ is called an outcome or an elementary event. The sample space can be finite, countably infinite, or uncountably infinite.

Events

An event is a subset $A \subseteq \Omega$ — a collection of outcomes. You say that event $A$ occurs if the actual outcome $\omega$ of the experiment belongs to $A$, i.e.\ $\omega \in A$.

Example. For the die roll $\Omega = \{1,2,3,4,5,6\}$:

• “Roll an even number”: $A = \{2, 4, 6\}$.
• “Roll at least five”: $B = \{5, 6\}$.
• “Roll a one”: $C = \{1\}$ (an elementary event as a set).

The empty set $\emptyset$ is the impossible event — it can never occur. The whole sample space $\Omega$ is the certain event — it always occurs.

Set operations on events

Set operations translate directly into logical operations on events.

Complement

The complement $A^c = \Omega \setminus A$ is the event ”$A$ does not occur”. It contains all outcomes not in $A$.

$$A^c \coloneqq \{\omega \in \Omega : \omega \notin A\}.$$

Union

The union $A \cup B$ is the event ”$A$ or $B$ (or both) occurs”. It contains all outcomes in at least one of the two events.

$$A \cup B \coloneqq \{\omega \in \Omega : \omega \in A \text{ or } \omega \in B\}.$$

More generally, $\bigcup_{n=1}^\infty A_n$ is the event that at least one of $A_1, A_2, \ldots$ occurs.

Intersection

The intersection $A \cap B$ is the event ”$A$ and $B$ both occur”. It contains outcomes common to both.

$$A \cap B \coloneqq \{\omega \in \Omega : \omega \in A \text{ and } \omega \in B\}.$$

If $A \cap B = \emptyset$, the events are mutually exclusive (disjoint): they cannot both occur in the same experiment.

Difference and symmetric difference

The difference $A \setminus B = A \cap B^c$ is the event ”$A$ occurs but $B$ does not”. The symmetric difference $A \triangle B = (A \setminus B) \cup (B \setminus A)$ is the event “exactly one of $A$, $B$ occurs”.

De Morgan’s laws

De Morgan’s laws translate between complement–union and complement–intersection:

$$(A \cup B)^c = A^c \cap B^c, \qquad (A \cap B)^c = A^c \cup B^c.$$

In words: “neither $A$ nor $B$” is the same as “not $A$ and not $B$”, and “not both $A$ and $B$” is the same as “not $A$ or not $B$”. These identities extend to countable collections:

$$\Bigl(\bigcup_{n=1}^\infty A_n\Bigr)^c = \bigcap_{n=1}^\infty A_n^c, \qquad \Bigl(\bigcap_{n=1}^\infty A_n\Bigr)^c = \bigcup_{n=1}^\infty A_n^c.$$

Sequences of events: limsup and liminf

For an infinite sequence of events $A_1, A_2, \ldots$, two derived events capture long-run behaviour:

$$\limsup_{n\to\infty} A_n \coloneqq \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k$$

is the event ”$A_n$ occurs infinitely often” (abbreviated i.o.): for every $n$, some $A_k$ with $k \geq n$ occurs.

$$\liminf_{n\to\infty} A_n \coloneqq \bigcup_{n=1}^\infty \bigcap_{k=n}^\infty A_k$$

is the event ”$A_n$ occurs all but finitely often”: from some point on, every $A_k$ occurs. The inclusion $\liminf A_n \subseteq \limsup A_n$ always holds.

These set-theoretic definitions are the foundation of the Borel–Cantelli lemmas, which translate convergence of $\sum_n P(A_n)$ into almost-sure results about whether infinitely many $A_n$ occur.

Why not every subset can be an event

For a finite or countably infinite sample space, you can safely declare every subset of $\Omega$ an event. For uncountable sample spaces such as $\Omega = \mathbb{R}$, assigning consistent probabilities to all $2^{|\mathbb{R}|}$ subsets leads to a contradiction (Vitali’s theorem shows such subsets exist). The resolution is to restrict attention to a $\sigma$-algebra $\mathcal{F} \subseteq 2^\Omega$ — a collection of subsets closed under complement and countable union — and only call elements of $\mathcal{F}$ events. The full treatment of $\sigma$-algebras is in The Probability Axioms.

Summary

• The sample space $\Omega$ is the set of all possible outcomes; each $\omega \in \Omega$ is an outcome.
• An event is a subset $A \subseteq \Omega$; event $A$ occurs if the outcome $\omega \in A$.
• The empty set $\emptyset$ is the impossible event; $\Omega$ is the certain event; two events with $A \cap B = \emptyset$ are mutually exclusive.
• Complement ($A^c$), union ($A \cup B$), and intersection ($A \cap B$) correspond to “not”, “or”, and “and” in logic.
• De Morgan’s laws convert between unions and intersections under complementation.
• The limsup of a sequence of events is the event that infinitely many of them occur; the liminf is the event that all but finitely many occur.
• For uncountable $\Omega$, not every subset can be an event — only members of a chosen $\sigma$-algebra $\mathcal{F}$.