Chapter 1A: Why Probability? Sample Spaces and Events

MATH/STAT 394: Probability I

Arman Jahangiri

University of Washington Department of Mathematics

Summer 2026

Arman Jahangiri

Chapter 1A: Why Probability? Sample Spaces and Events

Summer 2026

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Learning Goals

By the end of this lecture, students should be able to:

▶ explain probability as a mathematical language for uncertainty;

▶ identify a sample space for a random experiment;

▶ distinguish between outcomes and events;

▶ translate basic probability statements into set notation;

▶ use unions, intersections, complements, and De Morgan’s laws;

▶ recognize why choosing the correct sample space matters.

Arman Jahangiri

Chapter 1A: Why Probability? Sample Spaces and Events

Summer 2026

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Why Study Probability?

Central idea

Mathematics is the logic of certainty. Probability is the logic of uncertainty.

Probability provides tools for:

▶ modeling randomness and uncertainty;

▶ understanding variation and separating signal from noise;

▶ making principled predictions and decisions;

▶ quantifying risk;

▶ avoiding misleading or incomplete intuition.

Instructor note / optional discussion

Ask students for examples of uncertainty from everyday life: weather, medical tests, elections, gambling,

machine learning, stock prices, sports, or exams.

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Chapter 1A: Why Probability? Sample Spaces and Events

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Three Habits for This Course

1

Biohazards. Study common mistakes so that invalid reasoning becomes easier to detect.

2

Sanity checks. After solving a problem, check whether the answer makes sense in simple or extreme cases.

3

Conceptual clarity. Make sure the sample space, events, and assumptions are clearly speciﬁed before

assigning probabilities.

Guiding principle

Good probability reasoning starts with precise modeling, not just calculations.

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Chapter 1A: Why Probability? Sample Spaces and Events

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Sample Spaces and Events

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Chapter 1A: Why Probability? Sample Spaces and Events

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Random Experiments

A random experiment is a process whose outcome is uncertain before it is performed.

Examples:

▶ ﬂipping a coin;

▶ rolling a die;

▶ drawing a card from a shuﬄed deck;

▶ recording tomorrow’s temperature;

▶ selecting a student from the class;

▶ measuring the lifetime of a machine component.

Important

Before assigning probabilities, the possible outcomes must be clearly speciﬁed.

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Chapter 1A: Why Probability? Sample Spaces and Events

Summer 2026

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Deﬁnition: Sample Space and Event

Deﬁnition

The sample space S of an experiment is the set of all possible outcomes.

An event A is a subset of the sample space:

A ⊆ S.

The event A occurs if the actual outcome belongs to A:

s_actual∈ A.

▶ An outcome is a single element of S.

▶ An event is a collection of outcomes.

▶ The sample space is the universal set for the experiment.

▶ Some textbooks use Ω instead of S, and ω instead of s

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Chapter 1A: Why Probability? Sample Spaces and Events

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Pebble World Picture

When the sample space is ﬁnite, imagine each possible outcome as a pebble.

S = {all pebbles}

S

A

B

▶ Each pebble represents an outcome.

▶ An event is a set of pebbles.

▶ Performing the experiment selects one pebble.

▶ The event occurs if the selected pebble lies in the event.

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Chapter 1A: Why Probability? Sample Spaces and Events

Summer 2026

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Types of Sample Spaces

Sample spaces can be:

1

Finite

S = {1, 2, 3, 4, 5, 6}

S = {1, 2, 3, . . . }

S = [0, ∞)

for rolling one die.

2

Countably inﬁnite

for the number of tosses until the ﬁrst head.

3

Uncountable

for the lifetime of a machine component.

For now

Many early examples use ﬁnite sample spaces, where outcomes can be counted.

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Chapter 1A: Why Probability? Sample Spaces and Events

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Set Operations for Events

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Union, Intersection, Complement

Let A, B ⊆ S be events.

Set operation Notation

Meaning

Union

Intersection

Complement

A ∪ B

A or B occurs, possibly both

both A and B occur

A ∩ B

A^c(or A^′) A does not occur

Notation

Some textbooks write A^′for the complement of A. In this course, we write A^cinstead, because prime

notation is also used for diﬀerentiation, such as f ^′(x).

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Venn Diagrams for Basic Set Operations

The green area represents the formula above each diagram. Source: ResearchGate Figure

Here Ω represent the sample space (S) and A^′represent complement of A, i.e., A^c

Chapter 1A: Why Probability? Sample Spaces and Events

.

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Example: Ten Coin Flips

Suppose a coin is ﬂipped 10 times. Encode Heads as 1 and Tails as 0.

S = {(s₁, . . . , s₁₀) : s_j∈ {0, 1}, 1 ≤ j ≤ 10}.

There are

|S| = 2¹⁰

possible outcomes.

Let A_jbe the event that the jth ﬂip is heads:

A_j= {(s₁, . . . , s₁₀) ∈ S : s_j= 1}.

10

[

At least one head =

All ﬂips are heads =

A_j.

j=1

10

\

A_j.

j=1

9

[

Chapter 1A: Why Probability? Sample Spaces and Events

At least two consecutive heads =

(A_j∩ A_j+1).

j=1

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The Empty Set

The empty set is the set with no elements. It is denoted by

∅.

Important facts:

▶ ∅ means “nothing occurred” or “no outcome is included.”

▶ For every set A,

A ∩ ∅ = ∅.

▶ For every set A,

A ∪ ∅ = A.

▶ The empty set is a subset of every set:

∅ ⊆ A.

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Important Set Notation Clariﬁcations

Singletons versus elements

The object 2 is not the same as the set {2}.

2

̸

The set {2} is the set whose only element is 2.

Similarly,

∅

̸

▶ ∅ has no elements.

▶ {∅} has one element, namely ∅.

So:

|∅| = 0,

|{∅}| = 1.

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Events as Sets: A versus {A}

Suppose A is an event. Then A is already a set of outcomes.

For example, if

S = {1, 2, 3, 4, 5, 6},

A = {2, 4, 6},

then A is the event that the die roll is even.

But

ꢀ

{A} = {2, 4, 6}

is a set whose single element is the set A.

Common mistake

Usually, A ⊆ S, but {A} is not a subset of S unless A itself is an outcome.

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De Morgan’s Laws

For events A, B ⊆ S,

(A ∪ B)^c= A^c∩ B^c,

(A ∩ B)^c= A^c∪ B^c.

Interpretation:

▶ Not having at least one of A or B means having neither.

▶ Not having both A and B means at least one fails to occur.

Optional live veriﬁcation

Draw a Venn diagram and shade both sides of each identity.

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Union and Intersection of Many Events

Let A₁, A₂, . . . , A_n⊆ S.

The union is the event that at least one of the events occurs:

n

[

A_i= {s ∈ S : s ∈ A_ifor at least one i ∈ {1, . . . , n}}.

i=1

The intersection is the event that all of the events occur:

n

\

A_i= {s ∈ S : s ∈ A_ifor every i ∈ {1, . . . , n}}.

i=1

For inﬁnitely many events A₁, A₂, . . . :

∞

[

\

A_i= {s ∈ S : s ∈ A_ifor at least one i},

A_i= {s ∈ S : s ∈ A_ifor every i}.

i=1

Question: What does demorgan law generalize to, for a union of (inﬁnitely many) events?

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Example: Drawing One Card

Draw one card from a standard 52-card deck.

Deﬁne:

A = {card is an ace},

B = {card has a black suit},

D = {card is a diamond},

H = {card is a heart}.

Then:

A ∩ H = {Ace of Hearts},

A ∩ B = {Ace of Spades, Ace of Clubs},

D ∩ H = ∅,

because no card can be both a diamond and a heart.

A ∪ D ∪ H = {card is red or an ace}.

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Card Example: Complements

Using De Morgan’s laws,

(A ∪ B)^c= A^c∩ B^c.

(A ∪ B)^c= {card is not an ace and not black}.

(A ∪ B)^c= {card is a red non-ace}.

(D ∪ H)^c= D^c∩ H^c= B.

So:

Equivalently,

Also,

Lesson

The same event can often be expressed in more than one way. Some expressions are much easier to use than

others.

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A Subtle Point: Is the Sample Space Correct?

In the card example, the sample space is the set of all 52 standard cards.

If the drawn card is a joker, then the issue is not probability calculation.

The issue is:

the sample space was wrong.

Rule

The sample space must contain every outcome that can actually occur.

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English ←→ Set Notation

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Dictionary: Events and Occurrences

Let s_actualbe the actual outcome.

English statement

Set notation

sample space

s is a possible outcome

A is an event

A occurred

something must happen s_actual∈ S

S

s ∈ S

A ⊆ S

s_actual∈ A

Conceptual point

Events are not numbers. Events are sets. Probability will later assign numbers to events.

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Dictionary: New Events from Old Events

English statement

Set notation

A or B

A and B

not A

A ∪ B

A ∩ B

A^c

A or B, but not both

(A ∩ B^c) ∪ (A^c∩ B)

n

[

at least one of A₁, . . . , A_n

A_j

j=1

n

\

all of A₁, . . . , A_n

A_j

j=1

Notation reminder

We write A^cfor complement, not A^′.

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Dictionary: Relationships Between Events

English statement

Set notation

A implies B

A and B are mutually exclusive

A ⊆ B

A ∩ B = ∅

A₁∪ · · · ∪ A_n= S,

A₁, . . . , A_nform a partition of S

A_i∩ A_j= ∅ for i

̸= j

Partition

A partition divides the sample space into non-overlapping pieces that together cover all possible outcomes.

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Mini-Exercise: Translate to Set Notation

Let A_jbe the event that the jth coin ﬂip is heads.

Translate each statement:

1

The ﬁrst ﬂip is tails.

2

The ﬁrst two ﬂips are heads.

3

At least one of the ﬁrst three ﬂips is heads.

4

None of the ﬁrst three ﬂips is heads.

5

Exactly one of the ﬁrst two ﬂips is heads.

Optional answers

A^c₁,

A₁∩ A₂,

A₁∪ A₂∪ A₃,

(A₁∪ A₂∪ A₃)^c= A^c₁∩ A^c₂∩ A^c₃,

(A₁∩ A^c₂) ∪ (A^c₁∩ A₂).

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Chapter 1A: Why Probability? Sample Spaces and Events

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Common Mistake

Biohazard

Do not confuse an outcome with an event.

For one die roll:

S = {1, 2, 3, 4, 5, 6}.

The number 4 is an outcome.

The event “the roll is even” is the set

E = {2, 4, 6}.

So:

4 ∈ S,

E ⊆ S.

Optional emphasis

Writing E ∈ S is wrong here because E is a set of outcomes, not a single outcome.

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Summary

▶ Probability begins with a clearly speciﬁed sample space S.

▶ Outcomes are elements of S.

▶ Events are subsets of S.

▶ An event occurs when the actual outcome lies in that event.

▶ Set operations allow us to build complicated events from simpler events.

▶ De Morgan’s laws are essential tools for rewriting complements.

▶ Choosing the correct sample space is part of the modeling process.

Next lecture

Naive probability: when outcomes are equally likely, probability becomes counting.

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Chapter 1A: Why Probability? Sample Spaces and Events

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