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

Result R4847 on D5811: Event-conditional probability

Probability of event conditional on itself

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 $P$
(ii)	\begin{equation} \mathbb{P}(E) > 0 \end{equation}

Then \begin{equation} \mathbb{P}(E \mid E) = 1 \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 $P$
(ii)	\begin{equation} \mathbb{P}(E) > 0 \end{equation}

Proceeding from the definition, we have \begin{equation} \mathbb{P}(E \mid E) = \frac{\mathbb{P}(E \cap E)}{\mathbb{P}(E)} = \frac{\mathbb{P}(E)}{\mathbb{P}(E)} = 1 \end{equation} $\square$