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

Result R4073 on D1159: Probability space

Probability of union with the impossible event

Formulation 0

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$E \in \mathcal{F}$ is an D1716: Event in $P$

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

Formulation 1

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$E \in \mathcal{F}$ is an D1716: Event in $P$

Then \begin{equation} \mathbb{P}(E \text{ or } \emptyset) = \mathbb{P}(E) \end{equation}

Formulation 2

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$E \in \mathcal{F}$ is an D1716: Event in $P$

Then \begin{equation} \mathbb{P}(E \lor \emptyset) = \mathbb{P}(E) \end{equation}

Proofs

Proof 0

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$E \in \mathcal{F}$ is an D1716: Event in $P$

Result R2082: Binary set union with empty set shows that $E \cup \emptyset = E$, whence the claim follows. $\square$