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) $E, F_1, \ldots, F_N \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) $F_1, \ldots, F_N$ is a D5143: Set partition of $\Omega$
(iii) \begin{equation} \mathbb{P}(F_1), \ldots, \mathbb{P}(F_N) > 0 \end{equation}
Then \begin{equation} \mathbb{P}(E) = \sum_{n = 1}^N \mathbb{P}(E \mid F_n) \mathbb{P}(F_n) \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E, F_1, \ldots, F_N \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) $F_1, \ldots, F_N$ is a D5143: Set partition of $\Omega$
(iii) \begin{equation} \mathbb{P}(F_1), \ldots, \mathbb{P}(F_N) > 0 \end{equation}