ThmDex – An index of mathematical definitions, results, and conjectures.
Probability of complement of a null 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) = 0 \end{equation}
Then \begin{equation} \mathbb{P}(\Omega \setminus E) = 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 $P$
(ii) \begin{equation} \mathbb{P}(E) = 0 \end{equation}
Then \begin{equation} \mathbb{P}(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 $P$
(ii) \begin{equation} \mathbb{P}(E) = 0 \end{equation}
This result is a particular case of R3719: Probability of complement event. $\square$