Conditional probability

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 basic real number $$\mathbb{P}(E \mid \mathcal{G}) := \mathbb{E}(I_E \mid \mathcal{G})$$

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 basic real number $$\Omega \to [0, 1], \quad \omega \mapsto \mathbb{E}(I_E \mid \mathcal{G})(\omega)$$
Also known as
Posterior probability
Child definitions
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