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 $P$

The conditional probability of $E$ in $P$ given $\mathcal{G}$ is the D3161: Random real number \begin{equation} \mathbb{P}(E \mid \mathcal{G}) := \mathbb{E}(I_E \mid \mathcal{G}) \end{equation}

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 $P$

The conditional probability of $E$ in $P$ given $\mathcal{G}$ is the D3161: Random real number \begin{equation} \Omega \to [0, 1], \quad \omega \mapsto \mathbb{E}(I_E \mid \mathcal{G})(\omega) \end{equation}