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

\mathbb P(A\mid B)\;=\;\frac{\mathbb P(A\cap B)}{\mathbb P(B)}.

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

\mathbb P(A\mid \mathcal G)\;=\;\mathbb E[\mathbf 1_A\mid \mathcal G],

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

Remarks

Conditioning on $\sigma(B)=\{\varnothing,B,B^c,\Omega\}$ records both sides of the partition: when $0<\mathbb P(B)<1$ ,

\mathbb P(A\mid\sigma(B)) =\mathbb P(A\mid B)\mathbf 1_B+\mathbb P(A\mid B^c)\mathbf 1_{B^c} \quad\text{almost surely}.

Thus the event-conditioned number $\mathbb P(A\mid B)$ is the value of this random variable on $B$ , not the entire conditional random variable.

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 $\mathbb P(A\mid \mathcal G)=\mathbb P(A)$ .