Simple Events are Mutually Exclusive
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Theorem
Let $\EE$ be an experiment.
Let $e_1$ and $e_2$ be distinct simple events in $\EE$.
Then $e_1$ and $e_2$ are mutually exclusive.
Proof
By definition of simple event:
\(\ds e_1\) | \(=\) | \(\ds \set {s_1}\) | ||||||||||||
\(\ds e_2\) | \(=\) | \(\ds \set {s_2}\) |
for some elementary events $s_1$ and $s_2$ of $\EE$ such that $s_1 \ne s_2$.
It follows that:
\(\ds e_1 \cap e_2\) | \(=\) | \(\ds \set {s_1} \cap \set {s_2}\) | Definition of $e_1$ and $e_2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \O\) | Definition of Set Intersection |
The result follows by definition of mutually exclusive events.
$\blacksquare$
Sources
- 1965: A.M. Arthurs: Probability Theory ... (previous) ... (next): Chapter $2$: Probability and Discrete Sample Spaces: $2.2$ Sample spaces and events