# 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:

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