Definition:Event

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 can be argued that it is easier to understand what is meant.

Sources

• Geoffrey Grimmett: Probability: An Introduction (1986): $\S 1.2$