Events

An event is a subset of the sample space — a collection of outcomes that we assign probability to. We assign probabilities to events, not individual outcomes. This matters especially for continuous sample spaces (where individual outcomes have probability zero) and compound outcomes like “the sum of two dice is 7”.

The Language of Events

When an outcome ω occurs, we can ask whether an event A “happened”:

Example: Roll a die once. Sample space Ω = {1, 2, 3, 4, 5, 6}

• Event A = “roll an even number” = {2, 4, 6}
• If we roll a 4, then event A occurred (4 ∈ A)
• If we roll a 3, then event A did not occur (3 ∉ A)

Set Operations on Events

Since events are sets, we combine them using set operations:

Union (A ∪ B)

Outcomes in A or B (or both).

$$A \cup B = \{\omega : \omega \in A \text{ or } \omega \in B\}$$

Example:

• A = “first roll is 1” = {(1,1), (1,2), (1,3), (1,4)}
• B = “sum is odd” = {(1,2), (1,4), (2,1), (2,3), …}
• A ∪ B = “first roll is 1 OR sum is odd”

Intersection (A ∩ B)

Outcomes in A and B simultaneously.

$$A \cap B = \{\omega : \omega \in A \text{ and } \omega \in B\}$$

Example:

• A ∩ B = “first roll is 1 AND sum is odd” = {(1,2), (1,4)}

Complement (Aᶜ)

Outcomes not in A.

$$A^c = \{\omega : \omega \notin A\} = \Omega \setminus A$$

Example:

• A = “roll an even number” = {2, 4, 6}
• Aᶜ = “roll an odd number” = {1, 3, 5}

Disjoint Events

Two events are disjoint (or mutually exclusive) if they have no outcomes in common:

$$A \cap B = \emptyset$$

Example:

• A = “roll an even number” = {2, 4, 6}
• B = “roll an odd number” = {1, 3, 5}

A and B are disjoint — you cannot roll both even and odd simultaneously.

Visual: Disjoint Events

Ω = {1, 2, 3, 4, 5, 6}
A = {2, 4, 6}    ← even
B = {1, 3, 5}    ← odd
A ∩ B = ∅        ← no overlap

Examples of Events

Experiment: Roll a 4-sided die twice. Sample space has 16 outcomes.

De Morgan’s Laws

Useful identities for simplifying expressions:

$$(A \cup B)^c = A^c \cap B^c$$

$$(A \cap B)^c = A^c \cup B^c$$

In words:

• “Not (A or B)” = “not A and not B”
• “Not (A and B)” = “not A or not B”

Partition of the Sample Space

A partition splits the sample space into non-overlapping pieces where every outcome belongs to exactly one piece.

Think of it like sorting marbles into buckets — each marble goes in exactly one bucket, and no marble is left out.

Definition

Events $A_1, A_2, \ldots, A_n$ form a partition if:

1. No overlap: any two different events share no outcomes ($A_i \cap A_j = \emptyset$ when $i \neq j$)
2. Nothing left out: every outcome is in at least one event ($A_1 \cup A_2 \cup \cdots \cup A_n = \Omega$)

Here each $A_i$ is an event (a subset of the sample space). The subscript $i$ is just a number that tells you which event — $A_1$ is the first event, $A_2$ is the second, and so on. When we write $A_i \cap A_j = \emptyset$ for $i \neq j$, it means: pick any two different events from the partition, and they share no outcomes. For example, $A_1 \cap A_2 = \emptyset$, $A_1 \cap A_3 = \emptyset$, $A_2 \cap A_3 = \emptyset$, etc.

Example: Rolling a Die

Roll a six-sided die. Sample space: $\Omega = \{1, 2, 3, 4, 5, 6\}$.

Partition by value:

Check the two conditions:

• No overlap? {1,2} ∩ {3,4} = ∅, {1,2} ∩ {5,6} = ∅, {3,4} ∩ {5,6} = ∅ ✓
• Nothing left out? {1,2} ∪ {3,4} ∪ {5,6} = {1,2,3,4,5,6} = Ω ✓

Every possible roll is in exactly one group.

Why Partitions Matter

Partitions are the backbone of the Law of Total Probability (coming later): if you know the probability of something within each piece, you can compute the overall probability by adding them up.