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In probability, an event is a way of describing what we care about when an experiment is performed. The sample space (covered in the previous chapter) lists all possible outcomes; an event is a chosen collection of those outcomes.
If the sample space is $S$, then an event $E$ is any subset of $S$:
$$E \subseteq S.$$
For example, when rolling a fair six‑sided die:
- Sample space: $S = \{1,2,3,4,5,6\}$ (all possible results)
- Event “rolling an even number”: $E = \{2,4,6\}$
- Event “rolling a number greater than 4”: $F = \{5,6\}$
In words: an event is “the thing you’re asking about,” represented as a set of outcomes.
Types of events
Although all events are just subsets of the sample space, it is useful to give names to some common kinds.
Simple (elementary) events
A simple event (or elementary event) is an event that contains exactly one outcome.
Example: roll a die.
- Event “roll a 4”: $E = \{4\}$ is simple.
- Event “roll an even number”: $\{2,4,6\}$ is not simple (it has three outcomes).
Simple events are useful because in many basic settings each simple event is assumed to have equal probability.
Compound events
A compound event has more than one outcome.
Example: flip two coins.
- Sample space (as ordered pairs): $S = \{HH, HT, TH, TT\}$.
- Event “exactly one head”: $E = \{HT, TH\}$ is a compound event.
- Event “two heads”: $F = \{HH\}$ is simple.
Most real questions involve compound events (“at least two successes,” “more than 5,” “someone arrives before 3 pm,” etc.).
Certain and impossible events
Some events always happen or can never happen, based on how the experiment is defined.
- The certain event (or sure event) is the event that always occurs. As a set, it is the entire sample space $S$.
- Example: when you roll a fair die, the event “you get a number from 1 to 6” is certain; it is just $S$ itself.
- The impossible event is the event that cannot occur. As a set, it is the empty set $\varnothing$.
- Example: when rolling a standard die, the event “roll a 7” is impossible: $\varnothing$.
In probability notation, later we will have:
- $P(S) = 1$ (certain event)
- $P(\varnothing) = 0$ (impossible event)
Complementary events
For an event $E$, the complement $E^c$ (sometimes written $\overline{E}$) is the event that $E$ does not occur.
Formally, if $S$ is the sample space,
$$E^c = \{\,\omega \in S : \omega \notin E\,\} = S \setminus E.$$
Examples:
- Flip a coin once. $S = \{H,T\}$.
- Let $E =$ “get heads” $= \{H\}$.
- Then $E^c =$ “not heads” $= \{T\}$.
- Roll a die. $S = \{1,2,3,4,5,6\}$.
- Let $E =$ “roll an even number” $= \{2,4,6\}$.
- Then $E^c =$ “roll an odd number” $= \{1,3,5\}$.
An event and its complement never occur together, and exactly one of them occurs in each trial.
Mutually exclusive (disjoint) events
Two events $A$ and $B$ are mutually exclusive or disjoint if they cannot both happen on the same trial.
In set terms:
$$A \cap B = \varnothing.$$
Examples:
- Roll a die:
- $A =$ “roll a 1 or 2” $= \{1,2\}$,
- $B =$ “roll a 5 or 6” $= \{5,6\}$.
- Here $A \cap B = \varnothing$, so they are mutually exclusive.
- But if
- $C =$ “roll an even number” $= \{2,4,6\}$,
- $D =$ “roll a number $>3$” $= \{4,5,6\}$,
- then $C \cap D = \{4,6\} \neq \varnothing$, so $C$ and $D$ are not mutually exclusive: sometimes both happen (for outcomes 4 or 6).
Mutual exclusivity is about “cannot occur together” in a single trial of the experiment.
Exhaustive events
A collection of events is exhaustive if together they cover the entire sample space.
Events $A_1, A_2, \dots, A_n$ are exhaustive if
$$A_1 \cup A_2 \cup \dots \cup A_n = S.$$
Example: roll a die.
- $A =$ “1 or 2” $= \{1,2\}$
- $B =$ “3 or 4” $= \{3,4\}$
- $C =$ “5 or 6” $= \{5,6\}$
Then $A,B,C$ are exhaustive, because $A \cup B \cup C = \{1,2,3,4,5,6\} = S$.
If, in addition, they are pairwise mutually exclusive (no overlaps), they form a partition of the sample space.
Events as sets: building new events
Since events are sets of outcomes, we can use basic set operations (with matching event language). The underlying set theory is treated elsewhere; here the focus is on how these operations look in probability.
Let $A$ and $B$ be events in the same sample space $S$.
Union of events: “$A$ or $B$”
The union $A \cup B$ is the event that at least one of $A$ or $B$ occurs.
Set definition:
$$A \cup B = \{\,\omega \in S : \omega \in A \text{ or } \omega \in B\,\}.$$
Interpretation:
- “$A$ or $B$ or both.”
Example: roll a die.
- $A =$ “even” $= \{2,4,6\}$
- $B =$ “greater than 3” $= \{4,5,6\}$
Then
$$A \cup B = \{2,4,5,6\},$$
the event “even or greater than 3 (or both).”
Intersection of events: “$A$ and $B$”
The intersection $A \cap B$ is the event that both $A$ and $B$ occur.
Set definition:
$$A \cap B = \{\,\omega \in S : \omega \in A \text{ and } \omega \in B\,\}.$$
Example: same die roll as above:
- $A =$ “even” $= \{2,4,6\}$
- $B =$ “greater than 3” $= \{4,5,6\}$
Then
$$A \cap B = \{4,6\},$$
the event “even and greater than 3.”
If $A$ and $B$ are mutually exclusive, then $A \cap B = \varnothing$ and there is no outcome where both happen together.
Complement: “not $A$”
Already introduced, but in set language:
- $A^c = S \setminus A$, the event “$A$ does not occur.”
- It satisfies:
- $A \cup A^c = S$
- $A \cap A^c = \varnothing$
These relationships are the event version of basic set identities.
Events in finite and infinite sample spaces
Events work the same way whether the sample space is finite or infinite; the only change is how the subset looks.
Finite sample spaces
In a finite sample space, you can (at least in principle) list all outcomes.
Example: toss two coins.
- $S = \{HH, HT, TH, TT\}$
- Event “at least one head”: $A = \{HH, HT, TH\}$
- Event “exactly one head”: $B = \{HT, TH\}$
Here, describing an event is just listing the outcomes (or describing a rule that picks them).
Infinite sample spaces
In an infinite sample space, you cannot list all outcomes, but you can still define events by a rule.
Example: choose a real number at random from the interval $[0,1]$. Then:
- Sample space: $S = [0,1]$
- Event “number less than $0.5$”:
$$A = [0, 0.5)$$ - Event “number between $0.25$ and $0.75$”:
$$B = [0.25, 0.75]$$ - Event “number is exactly $0.3$”:
$$C = \{0.3\}$$
Each is still a subset of $S$, so each is an event.
Describing events in words, sets, and notation
In practice, events can be described in different but equivalent ways:
- In words: “at least two heads,” “sum is greater than 7,” “temperature above 30°C.”
- As a set of outcomes (explicit or rule-based).
- Using predicates/conditions on outcomes, for example:
- $A = \{ \omega \in S : X(\omega) \geq 3 \}$,
where $X$ is some numerical quantity associated with the outcome.
As problems become more involved, being able to move between these descriptions is important:
- Start from a verbal description,
- identify the underlying experiment and sample space,
- then identify the event as a precise subset of that space.
This chapter’s main idea is that events are the set-theoretic objects to which probabilities will be assigned. Once events are clearly defined, probabilities of those events can be meaningfully discussed in the later chapters.