The Central Idea
A sample space describes all modeled outcomes of an experiment. An event selects some of those outcomes.
If \(S\) is the sample space and \(A\) is an event, then
\[ \boxed{A\subseteq S}. \]Probability questions are usually questions about events rather than about isolated outcomes.
What Is an Event?
An event is any subset of the sample space.
Suppose a six-sided die is rolled, so
\[ S=\{1,2,3,4,5,6\}. \]The event “roll an even number” is
\[ \boxed{A=\{2,4,6\}}. \]The event occurs whenever the observed outcome belongs to \(A\).
Events Are Sets of Outcomes
Because events are sets, we can use ordinary set notation.
If
\[ A=\{2,4,6\}, \]then
\[ 4\in A \]means that outcome \(4\) makes the event occur, while
\[ 5\notin A \]means that outcome \(5\) does not.
Simple and Compound Events
A simple event contains exactly one outcome.
For example,
\[ \{4\} \]is the event that a die roll equals \(4\).
A compound event contains more than one outcome.
For example,
\[ \{2,4,6\} \]is the event that the die roll is even.
The Entire Sample Space Is an Event
Because every set is a subset of itself,
\[ S\subseteq S. \]Therefore, the entire sample space is an event.
This event represents a result that is certain to occur within the model.
\[ \boxed{\text{certain event}=S}. \]The Empty Set Is an Event
The empty set
\[ \varnothing \]is also a subset of every sample space.
It represents an impossible event: an event containing no outcomes.
\[ \boxed{\text{impossible event}=\varnothing}. \]Example: Events for One Die
Let
\[ S=\{1,2,3,4,5,6\}. \]Define
\[ A=\{\text{even outcomes}\}=\{2,4,6\}, \] \[ B=\{\text{outcomes greater than }3\}=\{4,5,6\}, \]and
\[ C=\{\text{outcomes equal to }1\}=\{1\}. \]Each event is a subset of the same sample space.
Event Notation Translates Language into Sets
Probability problems often begin with ordinary language.
For a die roll, the statement
“the result is less than \(4\)”
can be represented as
\[ \boxed{A=\{1,2,3\}}. \]Translating verbal conditions into subsets is a foundational probability skill.
The Union of Two Events
The union of events \(A\) and \(B\), written
\[ A\cup B, \]contains every outcome that belongs to \(A\), to \(B\), or to both.
\[ \boxed{A\cup B=\{x:x\in A\text{ or }x\in B\}}. \]In probability language, union usually corresponds to the word or.
Example: Union
Let
\[ A=\{2,4,6\} \]and
\[ B=\{4,5,6\}. \]Then
\[ \boxed{A\cup B=\{2,4,5,6\}}. \]The shared outcomes \(4\) and \(6\) are listed only once because a set does not repeat elements.
“Or” Is Usually Inclusive
In probability, the statement
“\(A\) or \(B\)”
normally means
\[ A\cup B, \]which includes outcomes belonging to both events.
Thus, probability uses the inclusive meaning of “or” unless the problem explicitly says “but not both” or otherwise excludes the overlap.
The Intersection of Two Events
The intersection of events \(A\) and \(B\), written
\[ A\cap B, \]contains the outcomes that belong to both events.
\[ \boxed{A\cap B=\{x:x\in A\text{ and }x\in B\}}. \]In probability language, intersection usually corresponds to and.
Example: Intersection
Let
\[ A=\{2,4,6\} \]and
\[ B=\{4,5,6\}. \]Then
\[ \boxed{A\cap B=\{4,6\}}. \]These are exactly the outcomes that satisfy both conditions.
The Complement of an Event
The complement of an event \(A\) consists of every outcome in the sample space that is not in \(A\).
Common notations include
\[ A^c \]and
\[ A'. \]We will use \(A^c\).
\[ \boxed{A^c=S\setminus A}. \]Example: Complement
Let
\[ S=\{1,2,3,4,5,6\} \]and
\[ A=\{2,4,6\}. \]Then
\[ \boxed{A^c=\{1,3,5\}}. \]If \(A\) is the event “roll an even number,” then \(A^c\) is the event “do not roll an even number,” which is the same as “roll an odd number.”
Set Difference
The difference
\[ A\setminus B \]contains the outcomes that are in \(A\) but not in \(B\).
\[ \boxed{A\setminus B=\{x:x\in A\text{ and }x\notin B\}}. \]This operation represents phrases such as “\(A\), but not \(B\)”.
Example: Difference
Let
\[ A=\{2,4,6\} \]and
\[ B=\{4,5,6\}. \]Then
\[ \boxed{A\setminus B=\{2\}}, \]while
\[ \boxed{B\setminus A=\{5\}}. \]In general,
\[ A\setminus B\neq B\setminus A. \]Difference as Intersection with a Complement
“In \(A\) but not in \(B\)” can also be written
\[ A\cap B^c. \]Therefore,
\[ \boxed{A\setminus B=A\cap B^c}. \]This identity is useful because it rewrites difference using intersection and complement.
Mutually Exclusive Events
Two events are mutually exclusive, or disjoint, if they cannot occur on the same trial.
In set notation,
\[ \boxed{A\cap B=\varnothing}. \]For a die roll, the events
\[ A=\{1,3,5\} \]and
\[ B=\{2,4,6\} \]are mutually exclusive.
Overlapping Events
Events overlap when they share at least one outcome.
If
\[ A\cap B\neq\varnothing, \]then \(A\) and \(B\) are not mutually exclusive.
For
\[ A=\{2,4,6\} \]and
\[ B=\{4,5,6\}, \]the overlap is
\[ A\cap B=\{4,6\}. \]Mutually Exclusive Does Not Mean Independent
Two concepts that must be kept separate are mutual exclusivity and independence.
Mutually exclusive events cannot occur together:
\[ A\cap B=\varnothing. \]Independence concerns whether the occurrence of one event changes the probability of another.
These ideas are different. Independence will be developed in a later lesson.
Two Important Complement Identities
An event and its complement together contain every outcome:
\[ \boxed{A\cup A^c=S}. \]They share no outcomes:
\[ \boxed{A\cap A^c=\varnothing}. \]Thus, \(A\) and \(A^c\) divide the sample space into two disjoint parts.
The Complement of the Complement
Taking the complement twice returns the original event:
\[ \boxed{(A^c)^c=A}. \]The first complement selects everything outside \(A\); the second complement selects everything outside that set, which returns to \(A\).
Complements of the Certain and Impossible Events
The complement of the entire sample space is the empty set:
\[ \boxed{S^c=\varnothing}. \]The complement of the empty set is the entire sample space:
\[ \boxed{\varnothing^c=S}. \]Union with the Empty Set
Adding no outcomes changes nothing:
\[ \boxed{A\cup\varnothing=A}. \]Likewise,
\[ \boxed{A\cap S=A}. \]These are basic identity laws for event operations.
Intersection with the Empty Set
No outcome can belong to both \(A\) and the empty set, so
\[ \boxed{A\cap\varnothing=\varnothing}. \]Similarly, because every outcome in \(A\) already belongs to the sample space,
\[ \boxed{A\cup S=S}. \]Commutative Laws
The order of events does not matter for union:
\[ \boxed{A\cup B=B\cup A}. \]The order also does not matter for intersection:
\[ \boxed{A\cap B=B\cap A}. \]Associative Laws
When several unions or intersections are combined, grouping does not change the result:
\[ \boxed{(A\cup B)\cup C=A\cup(B\cup C)}, \] \[ \boxed{(A\cap B)\cap C=A\cap(B\cap C)}. \]We can therefore write
\[ A\cup B\cup C \]or
\[ A\cap B\cap C \]without ambiguity.
Distributive Laws
Union and intersection distribute over one another:
\[ \boxed{A\cap(B\cup C)=(A\cap B)\cup(A\cap C)}, \] \[ \boxed{A\cup(B\cap C)=(A\cup B)\cap(A\cup C)}. \]These identities allow complicated event expressions to be reorganized without changing the outcomes they contain.
De Morgan's Laws
De Morgan's laws describe how complement interacts with union and intersection:
\[ \boxed{(A\cup B)^c=A^c\cap B^c}, \] \[ \boxed{(A\cap B)^c=A^c\cup B^c}. \]In words:
- not \(A\) or \(B\) means neither \(A\) nor \(B\);
- not both \(A\) and \(B\) means at least one of them does not occur.
Example: De Morgan's First Law
Suppose
\[ S=\{1,2,3,4,5,6\}, \] \[ A=\{2,4,6\}, \]and
\[ B=\{4,5,6\}. \]First,
\[ A\cup B=\{2,4,5,6\}. \]Therefore,
\[ (A\cup B)^c=\{1,3\}. \]Also,
\[ A^c=\{1,3,5\} \]and
\[ B^c=\{1,2,3\}. \]Hence,
\[ A^c\cap B^c=\{1,3\}. \]The two descriptions agree.
Example: De Morgan's Second Law
Using the same events,
\[ A\cap B=\{4,6\}, \]so
\[ (A\cap B)^c=\{1,2,3,5\}. \]Meanwhile,
\[ A^c\cup B^c = \{1,3,5\}\cup\{1,2,3\} = \{1,2,3,5\}. \]Therefore,
\[ \boxed{(A\cap B)^c=A^c\cup B^c}. \]“At Least One” as a Union
The phrase at least one often corresponds to a union.
If \(A\) means “the first component fails” and \(B\) means “the second component fails,” then
\[ A\cup B \]means that at least one component fails.
The overlap \(A\cap B\) is included because both failing still satisfies “at least one.”
“Both” as an Intersection
The phrase both \(A\) and \(B\) means
\[ \boxed{A\cap B}. \]For two dice, if \(A\) is “the first die is even” and \(B\) is “the second die is greater than \(4\),” then \(A\cap B\) contains exactly the outcomes satisfying both requirements.
“Neither” as a Complement
“Neither \(A\) nor \(B\)” means that neither event occurs.
This is the complement of their union:
\[ \boxed{(A\cup B)^c}. \]By De Morgan's law,
\[ \boxed{(A\cup B)^c=A^c\cap B^c}. \]“Not Both”
“Not both \(A\) and \(B\)” means that the two events do not occur together.
The corresponding event is
\[ \boxed{(A\cap B)^c}. \]By De Morgan's law,
\[ (A\cap B)^c=A^c\cup B^c. \]“Exactly One” of Two Events
“Exactly one of \(A\) and \(B\)” means that one occurs and the other does not.
This event can be written
\[ \boxed{(A\cap B^c)\cup(A^c\cap B)}. \]This operation is sometimes called the symmetric difference.
The Symmetric Difference
The symmetric difference of \(A\) and \(B\) contains outcomes belonging to exactly one of the two events.
A common notation is
\[ A\triangle B. \]Thus,
\[ \boxed{A\triangle B=(A\setminus B)\cup(B\setminus A)}. \]It can also be written
\[ \boxed{A\triangle B=(A\cup B)\setminus(A\cap B)}. \]“At Most One” of Two Events
“At most one of \(A\) and \(B\)” allows zero or one of them to occur, but not both.
Therefore,
\[ \boxed{\text{at most one of }A,B=(A\cap B)^c}. \]This includes the cases in which neither occurs.
“At Least Two” of Three Events
For events \(A\), \(B\), and \(C\), “at least two occur” can be represented by
\[ \boxed{(A\cap B)\cup(A\cap C)\cup(B\cap C)}. \]An outcome belonging to all three events is included automatically because it belongs to every pairwise intersection.
“Exactly Two” of Three Events
“Exactly two occur” must exclude the case in which all three occur.
One precise representation is
\[ \boxed{ (A\cap B\cap C^c) \cup (A\cap C\cap B^c) \cup (B\cap C\cap A^c) }. \]Each term describes one particular pair occurring while the third event does not.
Partitions of the Sample Space
A collection of events forms a partition of \(S\) when:
- the events are mutually exclusive;
- their union is the entire sample space.
For a die roll,
\[ A=\{1,3,5\} \]and
\[ B=\{2,4,6\} \]form a partition because
\[ A\cap B=\varnothing \]and
\[ A\cup B=S. \]Events in a Two-Coin Experiment
Let
\[ S=\{HH,HT,TH,TT\}. \]Define
\[ A=\{\text{at least one head}\} \]and
\[ B=\{\text{exactly one head}\}. \]Then
\[ A=\{HH,HT,TH\}, \]while
\[ B=\{HT,TH\}. \]Notice that
\[ B\subseteq A. \]Subset Relationships Between Events
If every outcome in event \(A\) also belongs to event \(B\), then
\[ A\subseteq B. \]In probability language, occurrence of \(A\) guarantees occurrence of \(B\).
For example, when two coins are tossed,
\[ \{\text{exactly two heads}\} \subseteq \{\text{at least one head}\}. \]Equivalent Event Descriptions
Different verbal statements can describe the same subset of the sample space.
For a die roll,
“the result is not even”
and
“the result is odd”
both describe
\[ \{1,3,5\}. \]Probability calculations depend on the event itself, not on which equivalent wording is used.
Events in a Two-Dice Sample Space
Suppose two distinguishable dice are rolled:
\[ S=\{(i,j):i,j\in\{1,2,3,4,5,6\}\}. \]Let
\[ A=\{(i,j):i+j=7\} \]and
\[ B=\{(i,j):i=j\}. \]Then \(A\) is the event “sum \(7\),” while \(B\) is the event “doubles.”
Example: Union for Two Dice
With
\[ A=\{(i,j):i+j=7\} \]and
\[ B=\{(i,j):i=j\}, \]the union
\[ A\cup B \]is the event
“the sum is \(7\) or doubles are rolled.”
Because no double has sum \(7\), these particular events are disjoint.
Example: Intersection for Two Dice
Let
\[ C=\{(i,j):i+j\text{ is even}\} \]and
\[ D=\{(i,j):i=j\}. \]Every double has an even sum, so
\[ D\subseteq C. \]Therefore,
\[ \boxed{C\cap D=D}. \]Event Operations with Continuous Sample Spaces
Event operations work the same way for continuous sample spaces.
Suppose
\[ S=[0,10]. \]Let
\[ A=[0,4] \]and
\[ B=[3,7]. \]Then
\[ \boxed{A\cap B=[3,4]} \]and
\[ \boxed{A\cup B=[0,7]}. \]Complement in a Continuous Sample Space
Continue with
\[ S=[0,10] \]and
\[ A=[0,4]. \]The complement must be taken relative to the sample space \(S\):
\[ \boxed{A^c=(4,10]}. \]It is not all real numbers outside \([0,4]\); it is only the portion of the sample space outside \(A\).
Complements Are Relative to the Sample Space
The meaning of \(A^c\) depends on \(S\).
If
\[ A=\{1,2\} \]and
\[ S=\{1,2,3,4\}, \]then
\[ A^c=\{3,4\}. \]If the sample space instead were
\[ S=\{1,2,3,4,5,6\}, \]then
\[ A^c=\{3,4,5,6\}. \]A complement is never interpreted without a surrounding universe of outcomes.
Venn Diagrams as Event Models
A Venn diagram represents the sample space as a region and events as overlapping subregions.
The diagram makes several operations visual:
- \(A\cup B\): all points in either event;
- \(A\cap B\): the overlap;
- \(A^c\): the portion of the sample space outside \(A\);
- \(A\setminus B\): the portion of \(A\) outside \(B\).
The diagram is a representation of set relationships, not a replacement for precise notation.
Event Algebra and Logical Language
Event operations parallel logical operations:
\[ \boxed{ \begin{aligned} A\cup B &\longleftrightarrow A\text{ or }B,\\ A\cap B &\longleftrightarrow A\text{ and }B,\\ A^c &\longleftrightarrow \text{not }A. \end{aligned} } \]This connection is why Boolean logic, set theory, and probability share many of the same algebraic laws.
From Outcome Conditions to Event Expressions
Consider a number \(x\) selected from the sample space
\[ S=\{1,2,\ldots,20\}. \]Let \(A\) be “\(x\) is even” and \(B\) be “\(x\) is divisible by \(3\).”
Then:
- \(A\cup B\): \(x\) is even or divisible by \(3\);
- \(A\cap B\): \(x\) is divisible by both \(2\) and \(3\);
- \(A^c\): \(x\) is odd;
- \(A\setminus B\): \(x\) is even but not divisible by \(3\).
Example: Building the Sets Explicitly
Continue with
\[ S=\{1,2,\ldots,20\}. \]Then
\[ A=\{2,4,6,8,10,12,14,16,18,20\} \]and
\[ B=\{3,6,9,12,15,18\}. \]Their intersection is
\[ \boxed{A\cap B=\{6,12,18\}}. \]These are the outcomes divisible by \(6\).
Checking an Event Expression by Membership
One way to verify an event identity is to test what membership means.
For example,
\[ x\in A\setminus B \]means
\[ x\in A \]and
\[ x\notin B. \]That is exactly the condition
\[ x\in A\cap B^c. \]Therefore,
\[ \boxed{A\setminus B=A\cap B^c}. \]Common Error: Treating an Event as One Outcome
The event “roll an even number” is not the single object “even.”
In the face-value sample space
\[ S=\{1,2,3,4,5,6\}, \]the event must be represented by the outcomes that satisfy the condition:
\[ \boxed{\{2,4,6\}}. \]Common Error: Confusing Union and Intersection
If \(A\) means “even” and \(B\) means “greater than \(3\),” then
\[ A\cup B \]means “even or greater than \(3\),” while
\[ A\cap B \]means “even and greater than \(3\).”
The words or and and must be translated carefully.
Common Error: Using Exclusive “Or” Automatically
In ordinary conversation, “or” is sometimes used to mean exactly one alternative.
In probability,
\[ A\cup B \]includes the overlap \(A\cap B\).
If a problem intends exactly one event, that condition must be expressed explicitly.
Common Error: Taking a Complement Outside the Sample Space
If
\[ S=\{1,2,3,4,5,6\} \]and
\[ A=\{1,2\}, \]then
\[ \boxed{A^c=\{3,4,5,6\}}. \]Numbers such as \(7\) and \(8\) are not added to the complement because they were never outcomes in the sample space.
Common Error: Confusing Disjoint and Complementary Events
Complementary events are always disjoint, but disjoint events need not be complements.
If
\[ A=\{1\} \]and
\[ B=\{2\} \]in
\[ S=\{1,2,3,4,5,6\}, \]then \(A\cap B=\varnothing\), so they are disjoint.
But
\[ A^c=\{2,3,4,5,6\}\neq B. \]Therefore they are not complementary.
Common Error: Forgetting the Overlap in a Union
Suppose
\[ A=\{1,2,3\} \]and
\[ B=\{3,4,5\}. \]Then
\[ A\cup B=\{1,2,3,4,5\}, \]not a list containing \(3\) twice.
The outcome \(3\) belongs to both events, but a set records membership, not multiplicity.
Common Error: Confusing “Not Both” with “Neither”
These are different conditions.
“Not both” is
\[ (A\cap B)^c, \]while “neither” is
\[ (A\cup B)^c. \]“Not both” allows one event to occur. “Neither” allows neither event to occur.
A Reliable Procedure for Translating Event Language
- Identify the sample space.
- Define each named event precisely.
- Translate “or” as union unless exclusivity is explicitly required.
- Translate “and” as intersection.
- Translate “not” as complement.
- Translate “but not” as set difference or intersection with a complement.
- Check whether “at least,” “at most,” “exactly,” or “neither” changes the structure.
- Verify the expression by describing which outcomes it contains.
Example: Full Translation Workflow
Suppose \(A\) means “a selected customer buys product \(A\)” and \(B\) means “the customer buys product \(B\).”
Then:
- buys at least one product: \(A\cup B\);
- buys both products: \(A\cap B\);
- buys product \(A\) but not \(B\): \(A\cap B^c\);
- buys neither product: \(A^c\cap B^c=(A\cup B)^c\);
- buys exactly one product: \((A\cap B^c)\cup(A^c\cap B)\).
Event notation compresses several verbal conditions into a small number of precise operations.
Practice 1: Identify an Event
A six-sided die is rolled. Let \(A\) be the event that the result is greater than \(4\).
Write \(A\) as a set.
Practice 2: Union
Let
\[ A=\{1,2,3\} \]and
\[ B=\{3,4,5\}. \]Find \(A\cup B\).
Practice 3: Intersection
Using
\[ A=\{1,2,3\} \]and
\[ B=\{3,4,5\}, \]find \(A\cap B\).
Practice 4: Complement
Let
\[ S=\{1,2,3,4,5,6\} \]and
\[ A=\{2,4,6\}. \]Find \(A^c\).
Practice 5: Difference
Let
\[ A=\{1,2,3,4\} \]and
\[ B=\{3,4,5,6\}. \]Find \(A\setminus B\).
Practice 6: Translate “Neither”
Write an event expression for:
Neither \(A\) nor \(B\) occurs.
Practice 7: Translate “Exactly One”
Write an event expression for:
Exactly one of \(A\) and \(B\) occurs.
Practice 8: Mutually Exclusive Events
Suppose
\[ A=\{1,3,5\} \]and
\[ B=\{2,4,6\}. \]Are \(A\) and \(B\) mutually exclusive? Explain using intersection.
Practice 9: De Morgan's Law
Rewrite
\[ (A\cup B)^c \]without placing the complement outside parentheses.
Practice 10: Two-Coin Events
Let
\[ S=\{HH,HT,TH,TT\}. \]Let \(A\) be the event “at least one head” and \(B\) the event “exactly one head.”
Write \(A\), \(B\), and \(A\cap B\).
Practice 11: Continuous Events
Let
\[ S=[0,12], \] \[ A=[0,5], \]and
\[ B=[4,9]. \]Find \(A\cap B\) and \(A\cup B\).
Practice 12: Identify the Error
A student says:
“If two events are mutually exclusive, then they must be complements.”
Explain why this is incorrect.
Practice Answers
Practice 1
\[ \boxed{A=\{5,6\}}. \]Practice 2
\[ \boxed{A\cup B=\{1,2,3,4,5\}}. \]Practice 3
\[ \boxed{A\cap B=\{3\}}. \]Practice 4
\[ \boxed{A^c=\{1,3,5\}}. \]Practice 5
\[ \boxed{A\setminus B=\{1,2\}}. \]Practice 6
\[ \boxed{(A\cup B)^c=A^c\cap B^c}. \]Practice 7
\[ \boxed{(A\cap B^c)\cup(A^c\cap B)}. \]Practice 8
Yes. They are mutually exclusive because
\[ \boxed{A\cap B=\varnothing}. \]Practice 9
\[ \boxed{(A\cup B)^c=A^c\cap B^c}. \]Practice 10
\[ A=\{HH,HT,TH\}, \] \[ B=\{HT,TH\}, \]so
\[ \boxed{A\cap B=\{HT,TH\}=B}. \]Practice 11
\[ \boxed{A\cap B=[4,5]} \]and
\[ \boxed{A\cup B=[0,9]}. \]Practice 12
Mutually exclusive events only require
\[ A\cap B=\varnothing. \]Complementary events require both
\[ A\cap B=\varnothing \]and
\[ A\cup B=S. \]Two events can therefore be disjoint without covering the entire sample space.
What to Remember
Events are subsets of the sample space:
\[ \boxed{A\subseteq S}. \]The core event operations are
\[ \boxed{ \begin{aligned} A\cup B &:\text{ }A\text{ or }B,\\ A\cap B &:\text{ }A\text{ and }B,\\ A^c &:\text{ not }A,\\ A\setminus B &:\text{ }A\text{ but not }B. \end{aligned} } \]Key relationships include
\[ A\cup A^c=S, \] \[ A\cap A^c=\varnothing, \] \[ (A\cup B)^c=A^c\cap B^c, \]and
\[ (A\cap B)^c=A^c\cup B^c. \]- Union uses the inclusive meaning of “or.”
- Intersection represents simultaneous occurrence.
- Complements are always taken relative to the sample space.
- Disjoint events have empty intersection.
- Complementary events are disjoint and together fill the sample space.
- Different verbal descriptions can define the same event.
The most important habit is: translate the words into sets before manipulating probabilities.
Where We Go Next
Events identify which outcomes matter, and event operations organize how those collections of outcomes relate.
The next lesson, Axioms of Probability, assigns numerical probabilities to events and develops the basic rules that every valid probability model must satisfy.