Conditional probability

A conditional probability is the probability of an event after restricting to information in another event or in a sub-sigma-algebra . On a probability space $(\Omega,\mathcal F,\mathbb P)$, for events $A,B\in\mathcal F$ with $\mathbb P(B)>0$, the conditional probability of $A$ given $B$ is

More generally, for a sub-$\sigma$-algebra $\mathcal G\subseteq\mathcal F$, the conditional probability of $A$ given $\mathcal G$ is the $\mathcal G$-measurable random variable defined by

where $\mathbb E[\cdot\mid \mathcal G]$ denotes conditional expectation .

Conditioning on an event $B$ can be viewed as conditioning on the $\sigma$-algebra $\sigma(B)=\{\varnothing,B,B^c,\Omega\}$; conditional probability is central to independence of events and Bayesian updating.

Examples:

• If a fair die is rolled, $A=\{\text{even}\}$ and $B=\{\text{roll}>3\}$, then $\mathbb P(A\mid B)=\frac{2/6}{3/6}=\frac{2}{3}$.
• If $A$ and $B$ are independent events with $\mathbb P(B)>0$, then $\mathbb P(A\mid B)=\mathbb P(A)$.
• For a sub-$\sigma$-algebra $\mathcal G$, if $A\in\mathcal G$ then $\mathbb P(A\mid \mathcal G)=\mathbf 1_A$ almost surely, while if $\mathcal G=\{\varnothing,\Omega\}$ then $\mathbb P(A\mid \mathcal G)=\mathbb P(A)$.