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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): event
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): event