ThmDex – An index of mathematical definitions, results, and conjectures.
Result R2164 on D1716: Event
Law of the excluded middle in probability calculus
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E \in \mathcal{F}$ is an D1716: Event in $P$
Then \begin{equation} \mathbb{P}(E \cup E^{\complement}) = 1 \end{equation}
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E \in \mathcal{F}$ is an D1716: Event in $P$
Then \begin{equation} \mathbb{P}(E \text{ or } E^{\complement}) = 1 \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E \in \mathcal{F}$ is an D1716: Event in $P$
Using result R977: Ambient set is union of subset and complement of subset, we have \begin{equation} \mathbb{P}(E \cup E^{\complement}) = \mathbb{P}(\Omega) = 1 \end{equation} $\square$
Proof 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E \in \mathcal{F}$ is an D1716: Event in $P$
Using results
(i) R5101: Disjoint additivity of probability measure for binary union
(ii) R2060: Probability of set difference
(iii) R3719: Probability of complement event

we have \begin{equation} \mathbb{P}(E \cup E^{\complement}) = \mathbb{P}(E) + \mathbb{P}(E^{\complement}) = \mathbb{P}(E) + (1 - \mathbb{P}(E)) = 1 \end{equation} $\square$