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

Result R4338 on D2012: Conditional probability

Conditional probability of almost surely true event

Formulation 0

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

(i)	$\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$
(ii)	$E \in \mathcal{F}$ is an D1716: Event in $P$
(iii)	\begin{equation} \mathbb{P}(E) = 1 \end{equation}

Then \begin{equation} \mathbb{P}(E \mid \mathcal{G}) \overset{a.s.}{=} 1 \end{equation}

Formulation 1

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

(i)	$\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$
(ii)	$E \in \mathcal{F}$ is an D1716: Event in $P$
(iii)	\begin{equation} \mathbb{P}(E) = 1 \end{equation}

Then \begin{equation} \mathbb{P}( \mathbb{P}(E \mid \mathcal{G}) = 1) = 1 \end{equation}

Subresults

▶	R4822: Conditional probability of the sample space

Proofs

Proof 0

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

(i)	$\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$
(ii)	$E \in \mathcal{F}$ is an D1716: Event in $P$
(iii)	\begin{equation} \mathbb{P}(E) = 1 \end{equation}

Result R3649: Expectation of conditional probability shows that \begin{equation} \mathbb{E}(\mathbb{P}(E \mid \mathcal{G})) = \mathbb{P}(E) = 1 \end{equation} Since the random basic real number $\mathbb{P}(E \mid \mathcal{G})$ takes values in the interval $[0, 1]$, result R4398: then guarantees that \begin{equation} \mathbb{P}( \mathbb{P}(E \mid \mathcal{G}) = 1) = 1 \end{equation} $\square$