ThmDex – An index of mathematical definitions, results, and conjectures.
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E, F, G \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) $F, G$ is a D5143: Set partition of $\Omega$
(iii) \begin{equation} \mathbb{P}(F), \mathbb{P}(G) > 0 \end{equation}
Then \begin{equation} \mathbb{P}(E) = \mathbb{P}(E \mid F) \mathbb{P}(F) + \mathbb{P}(E \mid G) \mathbb{P}(G) \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E, F, G \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) $F, G$ is a D5143: Set partition of $\Omega$
(iii) \begin{equation} \mathbb{P}(F), \mathbb{P}(G) > 0 \end{equation}
This result is a particular case of R4930: . $\square$