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} $\square$