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Definition D2012
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 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 real number $$\Omega \to [0, 1], \quad \omega \mapsto \mathbb{E}(I_E \mid \mathcal{G})(\omega)$$
Children
 ▶ Conditionally independent event collection
Results
 ▶ R4932 ▶ R4931 ▶ R4930 ▶ Bayes' theorem in the case of two pullback events ▶ Bayes' theorem in the case of event and complement ▶ Bayes' theorem in the case of two events ▶ Conditional probability given independent random variable ▶ Conditional probability given independent sigma-algebra ▶ Conditional probability of almost surely true event ▶ Conditional probability of complement event ▶ Conditional probability of the empty event ▶ Conditional probability of the sample space ▶ Expectation of conditional probability ▶ Law of total probability for a countable partition of events of positive probability ▶ Law of total probability for complement partition in terms of random variables ▶ Law of total probability for complex expectation in terms of pullback events ▶ Law of total probability for complex expectation in terms of pullback events of a discrete random variable ▶ Probability calculus expression for probability conditioned on event of nonzero probability