# Event Space of Experiment with Final Sample Space has Even Cardinality

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## Theorem

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

Let $\Omega$ be a finite set.

Then the event space $\Sigma$ consists of an even number of subsets of $\Omega$.

## Proof

Let $A \in \Sigma$ be one of the events of $\EE$.

We have by definition that $\Omega$ is itself an events of $\EE$.

Hence by Set Difference of Events is Event, $\Omega \setminus A$ is also an event of $\EE$.

As $A$ is arbitrary, the same applies to all events of $\EE$.

Thus all events of $\EE$ come in pairs: $A$ and $\Omega \setminus A$.

Hence the result.

$\blacksquare$

## Also see

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $1$: Events and probabilities: $1.2$: Outcomes and events: Exercise $5$