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

Result R4278 on D1159: Probability space

Expression for probability of event in terms 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}(E) = 1 - \mathbb{P}(E^{\complement}) \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) = 1 - \mathbb{P}(\Omega \setminus E) \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 $P$

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