Definition:Event/Occurrence/Intersection

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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 $\omega \in \Omega$.


Let $\omega \in A \cap B$, where $A \cap B$ denotes the intersection of $A$ and $B$.

Then both $A$ and $B$ occur.


Also denoted as

Some sources denote the occurrence of both $A$ and $B$ as $A B$.


Examples

Both Prime and Even

Consider the experiment $\EE$ such that $2$ (positive) integers are drawn at random from a table of random numbers.

Let $A$ be the event that at least $1$ of these integers is prime.

Let $B$ be the event that at least $1$ of these integers is even.


Then their intersections $A \cap B$ means:

at least one of the $2$ integers is even and at least one of the $2$ integers is prime.

That is, either:

one of the $2$ integers is $2$ (two)

or:

one of the $2$ integers is even and the other is prime.


Also see

  • Results about intersections of events can be found here.


Sources

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