ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4932 on D2012: Conditional probability
Subresult of R4931:
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E, F \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) \begin{equation} \mathbb{P}(F) \in (0, 1) \end{equation}
Then \begin{equation} \mathbb{P}(E) = \mathbb{P}(E \mid F) \mathbb{P}(F) + \mathbb{P}(E \mid F^{\complement}) \mathbb{P}(F^{\complement}) \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E, F \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) \begin{equation} \mathbb{P}(F) \in (0, 1) \end{equation}
This result is a particular case of R4931: . $\square$