# Definition:Event/Occurrence/Intersection

< Definition:Event | Occurrence(Redirected from Definition:Intersection of Events)

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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:

That is, either:

or:

## Also see

- Definition:Union of Events
- Definition:Difference of Events
- Definition:Symmetric Difference of Events

- Results about
**intersections of events**can be found**here**.

## Sources

- 1965: A.M. Arthurs:
*Probability Theory*... (previous) ... (next): Chapter $2$: Probability and Discrete Sample Spaces: $2.2$ Sample spaces and events - 1968: A.A. Sveshnikov:
*Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions*(translated by Richard A. Silverman) ... (previous) ... (next): $\text I$: Random Events: $1$. Relations among Random Events - 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $1$: Events and probabilities: $1.2$: Outcomes and events