# Definition:Complementary Event

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## Definition

Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.

Let $A \in \Sigma$ be an event in $\EE$.

The **complementary event** to $A$ is defined as $\relcomp \Omega A$.

That is, it is the subset of the sample space of $\EE$ consisting of all the elementary events of $\EE$ that are not in $A$.

## Also known as

The **complementary event** to $A$ is also referred to as the **opposite event**.

It can also be denoted $\overline A$.

## Also see

- Results about
**complementary 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