Independence of events - Imported Knowlpedia Content

Two events are independent if, on a probability space $(\Omega,\mathcal F,\mathbb P)$ , they satisfy

\mathbb P(A\cap B)\;=\;\mathbb P(A)\,\mathbb P(B).

A finite or countable family of events $(A_i)_{i\in I}$ is independent if for every finite subset $\{i_1,\dots,i_n\}\subseteq I$ ,

\mathbb P\!\left(\bigcap_{k=1}^n A_{i_k}\right)\;=\;\prod_{k=1}^n \mathbb P(A_{i_k}).

Independence can be expressed in terms of conditional probability: if $\mathbb P(B)>0$ , then $A$ and $B$ are independent exactly when $\mathbb P(A\mid B)=\mathbb P(A)$ . This notion extends from events to independence of sigma-algebras and to independence of random variables.

Examples

In two independent coin flips, let $A=\{\text{first flip is H}\}$ and $B=\{\text{second flip is H}\}$ . Then $\mathbb P(A)=\mathbb P(B)=1/2$ and $\mathbb P(A\cap B)=1/4$ , so $A$ and $B$ are independent.
For one fair die roll, let $A=\{\text{even}\}$ and $B=\{\text{roll}\le 3\}$ . Then $\mathbb P(A)=1/2$ , $\mathbb P(B)=1/2$ , but $\mathbb P(A\cap B)=\mathbb P(\{2\})=1/6\ne 1/4$ , so $A$ and $B$ are not independent.