ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4711 on D1716: Event
Probability of complement of event which is not the whole sample space need not be nonzero
Formulation 0
Let $P = (\{ 0, 1 \}, \mathcal{P} \{ 0, 1 \}, \mathbb{P})$ be a D5512: Single boolean trial probability space such that
(i) \begin{equation} \mathbb{P} \{ 1 \} = 1 \end{equation}
(ii) \begin{equation} \mathbb{P} \{ 0 \} = 0 \end{equation}
Then
(i) \begin{equation} \{ 1 \} \neq \{ 0, 1 \} \end{equation}
(ii) \begin{equation} \mathbb{P} \{ 1 \}^{\complement} = \mathbb{P} \{ 0 \} = 0 \end{equation}