Sample Space is Union of All Distinct Simple Events
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Theorem
Let $\EE$ be an experiment.
Let $\Omega$ denote the sample space of $\EE$.
Then $\Omega$ is the union of the set of simple events in $\EE$.
Proof
- $\Omega \subseteq \Omega$
That is, $\Omega$ is itself an event in $\EE$.
The result as an application of Non-Trivial Event is Union of Simple Events.
$\blacksquare$
Sources
- 1965: A.M. Arthurs: Probability Theory ... (previous) ... (next): Chapter $2$: Probability and Discrete Sample Spaces: $2.2$ Sample spaces and events