# Definition:Event/Occurrence

< Definition:Event(Redirected from Definition:Occurrence (Probability Theory))

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## Definition

Let the probability space of an experiment $\mathcal E$ be $\left({\Omega, \Sigma, \Pr}\right)$.

Let $A, B \in \Sigma$, 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 $\omega \notin A$, i.e $\omega \in \Omega \setminus A$, then
**$A$ does not occur**.

- If $\omega \in A \cup B$, then
**either $A$ or $B$ occur**.

- If $\omega \in A \cap B$, then
**both $A$ and $B$ occur**.

- If $\omega \in A \setminus B$, then
**$A$ occurs but $B$ does not occur**.

- If $\omega \in A * B$, where $*$ denotes symmetric difference, then
**either $A$ occurs or $B$ occurs, but not both**.

## 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.

## Sources

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