Definition:Disjoint Events

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

• Definition:Pairwise Disjoint Events

• 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): exclusive
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): mutually exclusive
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): exclusive
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): mutually exclusive events