Random experiments, sample spaces, and events
Distinguish outcomes from events and use unions, intersections, mutual exclusivity and complements in finite sample spaces.
Content updated: October 22, 2026
An outcome is what occurs in one trial; an event is a set of outcomes satisfying a condition. Set notation distinguishes “or,” “and,” and “does not occur” before probabilities are calculated.
Learning goals: construct a finite sample space; represent events, unions, intersections, mutually exclusive events, and complements.
Prerequisites: finite sets, elements, unions, and intersections.
On this page: Definition · Method · Examples · Practice · Further learning
1. Experiments, outcomes, and the sample space
A random experiment is an activity whose possible outcomes are specified by a model, while its actual outcome is unknown beforehand. In this lesson, the sample space $\Omega$ is a finite, nonempty set containing those outcomes. Each trial produces one element $\omega\in\Omega$.
An event $A$ is a subset of $\Omega$; it occurs precisely when the outcome $\omega$ belongs to $A$. Thus an outcome is not identified with every kind of event. In the finite model used here, an event containing just one outcome, $\{\omega\}$, is an elementary event.
The sure event is $\Omega$; the impossible event is $\varnothing$. Events that may or may not occur, depending on the outcome, are commonly called random events. Equally likely outcomes are not yet required; that assumption is needed only for computing probability by uniform counting.
2. Operations on events
The union $A\cup B$ occurs when at least one event occurs, including when both occur. The intersection $A\cap B$ occurs when both occur. Two events are mutually exclusive if $A\cap B=\varnothing$, meaning they cannot occur together in the same trial.
The complement $\overline A=\Omega\setminus A$ occurs when $A$ does not. Always $A\cup\overline A=\Omega$ and $A\cap\overline A=\varnothing$. Mutually exclusive events need not be complementary: their union must also be the entire sample space.
In a finite space with every subset allowed as an event, each event is the union of the elementary events corresponding to its elements. These elementary events are pairwise mutually exclusive. The empty union is the impossible event; the empty set is not an elementary event.
3. Twelve examples of events
Example 1. Tossing a coin
Toss a coin with tails and heads. Denote the outcomes by $S,N$, respectively, so $\Omega=\{S,N\}$. The tails event is $\{S\}$ and the heads event is $\{N\}$. Fairness is not needed to specify these sets.
Example 2. Rolling a six-sided die
Take $\Omega=\{1,2,3,4,5,6\}$, where the number records the dots shown. Set $A_j=\{j\}$ for $j=1,\ldots,6$. Let $A=\{2,4,6\}$ mean even, $B=\{1,3,5\}$ odd, $C=\{4,5,6\}$ at least 4 dots, and $D=\{1,2,3\}$ at most 3 dots. Let $E$ mean at least 1 dot and $F$ at least 7 dots.
Example 3. The sure event
Every outcome shows at least 1 dot, so $E=\Omega$. This follows from the specified model with exactly these six faces.
Example 4. The impossible event
No outcome shows at least 7 dots, so $F=\varnothing$. Mentioning 7 in the condition does not add it to the sample space.
Example 5. Nontrivial events
Each $A_j$, as well as $A,B,C,D$, is nonempty but differs from $\Omega$. Some outcomes make it occur, and others do not.
Example 6. Decomposition into elementary events
We have $A=A_2\cup A_4\cup A_6$, $B=A_1\cup A_3\cup A_5$, $C=A_4\cup A_5\cup A_6$, and $D=A_1\cup A_2\cup A_3$. Also, $E=\bigcup_{j=1}^{6}A_j$, whereas $F$ is the empty union. The even event has three favorable outcomes: 2,4,6.
Example 7. Unions of events
We have $A\cup B=C\cup D=\Omega$. Also, $A\cup C=\{2,4,5,6\}$ and $A\cup D=\{1,2,3,4,6\}$. An outcome belonging to both sets is still listed only once in the union.
Example 8. Intersections of events
We have $A\cap B=C\cap D=\varnothing$, $A\cap C=\{4,6\}$, and $B\cap D=\{1,3\}$. Retain only outcomes satisfying both conditions.
Example 9. Elementary events
The events $A_1,\ldots,A_6$ are elementary because each is a singleton. The event $A$ is not elementary: it contains three outcomes and is a union of singleton subsets. The event $F$ is not elementary because it is empty.
Example 10. Two mutually exclusive pairs
The events $A,B$ are mutually exclusive; so are $C,D$. The intersections calculated in Example 8 are empty. Different names or descriptions alone do not imply mutual exclusivity.
Example 11. The empty set and any event
For every $H\subseteq\Omega$, we have $H\cap\varnothing=\varnothing$. Thus the impossible event is mutually exclusive with every event, including itself.
Example 12. Complementary pairs
We have $\overline A=B$, $\overline C=D$, and $\overline E=F$. Each pair is both mutually exclusive and exhaustive, satisfying both conditions for complements.
4. Practice
- For the die experiment above, write the event $H$ of showing more than 4 dots and its complement.
- Calculate $A\cap D$ and $B\cup C$.
- Are the events $\{1\}$ and $\{2\}$ mutually exclusive? Are they complementary?
Hints and answers
- $H=\{5,6\}$, $\overline H=\{1,2,3,4\}$.
- $A\cap D=\{2\}$, $B\cup C=\{1,3,4,5,6\}$.
- They are mutually exclusive because their intersection is empty, but not complementary because their union $\{1,2\}$ is not $\Omega$.
5. Common mistakes
Do not confuse one outcome with an event containing several outcomes. “Or” in a union includes the possibility that both events occur. A complement depends on the sample space; check both union and intersection to establish that two events are complementary.
6. Further learning
Probability of an event (English lesson) · Conditional probability (English lesson) · The addition rule of probability (English lesson) · The multiplication rule of probability (English lesson).
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