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

Result R3719 on D1159: Probability space
Subresult of R2060: Probability of set difference

Probability of complement 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 $P$

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

\begin{equation} \mathbb{P}(E^{\complement}) = 1 - \mathbb{P}(E) \end{equation}

Subresults

▶	R4278: Expression for probability of event in terms of complement event
▶	R4560: Probability of complement of a null event
▶	R4559: Probability of complement of an almost sure event

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 $P$

By definition of a D198: Probability measure, one has $\mathbb{P}(\Omega) = 1$. Thus, this result is a particular case of R2060: Probability of set difference. $\square$