ThmDex – An index of mathematical definitions, results, and conjectures.

Result R5711 on D1159: Probability space
Subresult of R4445: Inclusion-exclusion principle for probability of binary union

Inclusion-exclusion principle for probability of binary union for disjoint sets

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} E \cap F = \emptyset \end{equation}

Then \begin{equation} \mathbb{P}(E \cup F) = \mathbb{P}(E) + \mathbb{P}(F) \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} E \cap F = \emptyset \end{equation}

Since $\mathbb{P}(E \cap F) = \mathbb{P}(\emptyset) = 0$, this result is a special case of R4445: Inclusion-exclusion principle for probability of binary union. $\square$