Definition:Event

Context
Probability Theory.

Definition
Let $$\mathcal E$$ be an experiment.

An event in $$\mathcal E$$ is an element of the event space $$\Sigma$$ of $$\mathcal E$$.

Occurrence
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.

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