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

Result R4559 on D1159: Probability space
Subresult of R3719: Probability of complement event

Probability of complement of an almost sure event

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 $P$
(ii)	\begin{equation} \mathbb{P}(E) = 1 \end{equation}

Then \begin{equation} \mathbb{P}(\Omega \setminus E) = 0 \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 $P$
(ii)	\begin{equation} \mathbb{P}(E) = 1 \end{equation}

Then \begin{equation} \mathbb{P}(E^{\complement}) = 0 \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 $P$
(ii)	\begin{equation} \mathbb{P}(E) = 1 \end{equation}

This result is a particular case of R3719: Probability of complement event. $\square$