Independence of events

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

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

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.