# Definition:Disjoint Events

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## Definition

Let $A$ and $B$ be events in a probability space.

Then $A$ and $B$ are **disjoint** if and only if:

- $A \cap B = \O$

It follows by definition of probability measure that $A$ and $B$ are **disjoint** if and only if:

- $\map \Pr {A \cap B} = 0$

That is, two events are **disjoint** if and only if the probability of them both occurring in the same experiment is zero.

That is, if and only if $A$ and $B$ can't happen together.

## Also known as

$A$ and $B$ are also referred to in this context as **mutually exclusive**.

## Also see

- Results about
**disjoint 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: $1.3$: Probabilities - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**mutually exclusive** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**mutually exclusive events**