# Definition:Event/Occurrence

## Definition

Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.

Let $A, B \in \Sigma$ be events, so that $A \subseteq \Omega$ and $B \subseteq \Omega$.

Let the outcome of the experiment be $\omega \in \Omega$.

Then the following real-world interpretations of the **occurrence** of events can be determined:

- If $\omega \in A$, then
**$A$ occurs**.

- If $\omega \notin A$, that is $\omega \in \Omega \setminus A$, then
**$A$ does not occur**.

### Union

Let $\omega \in A \cup B$, where $A \cup B$ denotes the union of $A$ and $B$.

Then **either $A$ or $B$ occur**.

### Intersection

Let $\omega \in A \cap B$, where $A \cap B$ denotes the intersection of $A$ and $B$.

Then **both $A$ and $B$ occur**.

### Difference

Let $\omega \in A \setminus B$, where $A \setminus B$ denotes the difference of $A$ and $B$.

Then **$A$ occurs but $B$ does not occur**.

### Symmetric Difference

Let $\omega \in A \symdif B$, where $A \symdif B$ denotes the symmetric difference of $A$ and $B$.

Then **either $A$ occurs or $B$ occurs, but not both**.

### Equality

Let $A, B \in \Sigma$ be events of $\EE$ such that $A = B$.

Then:

- the occurrence of $A$ inevitably brings about the occurrence of $B$

and:

- the occurrence of $B$ inevitably brings about the occurrence of $A$.

### Certainty

Let $A \in \Sigma$ be an event of $\EE$ whose probability of occurring is equal to $1$.

Then $A$ is described as **certain**.

That is, it is a **certainty** that $A$ occurs.

### Impossibility

Let $A \in \Sigma$ be an event of $\EE$ whose probability of occurring is equal to $0$.

Then $A$ is described as **impossible**.

That is, it is an **impossibility** for $A$ to occur.

## Also known as

The word **happen** is often used for **occur**, and it can be argued that it is easier to understand what is meant.

## Examples

### Electric Circuit 1

Consider the electric circuit:

Let event $A$ be that switch $A$ is open.

Let event $B_n$ for $n = 1, 2, 3$ be that switch $B_n$ is open.

Let $C$ be the event that no current flows from $M$ to $N$.

Then:

- $C = A \cup \paren {B_1 \cap B_2 \cap B_3}$

- $\overline C = \overline A \cap \paren {\overline {B_1} \cup \overline {B_2} \cup \overline {B_3} }$

## Also see

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $1$: Events and probabilities: $1.2$: Outcomes and events