ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4326 on D5500: Almost sure event

Probability of binary union with almost sure event

Formulation 0

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$E, F \in \mathcal{F}$ are each an D1716: Event in $P$
(ii)	\begin{equation} \mathbb{P}(E) = 1 \end{equation}

Then \begin{equation} \mathbb{P}(E \cup F) = 1 \end{equation}

Proofs

Proof 0

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$E, F \in \mathcal{F}$ are each an D1716: Event in $P$
(ii)	\begin{equation} \mathbb{P}(E) = 1 \end{equation}

This result is a particular case of R4328: Probability of finite union with an almost sure event. $\square$