independence (probability)

Independence is where probability theory diverges from classical measure theory. Independence is a strong condition that simplifies things a lot.

Definitions

For events

For a probability space $(\Omega ,\mathcal{F},P)$, events $A_1,A_2,\dots ,A_n$ are independent if for every sub-collection

$$P\left(\bigcap _{j=1}^k{A}_{i_j}\right)=\prod _{j=1}^{k}P(A_{i_j})$$

For collection of events

For a collection of events ${\mathcal{C}}_1,{\mathcal{C}}_2,\dots$ (i.e. ${\mathcal{C}}_i\subset \mathcal{F}$) the collections are independent if for every choice $C_i\in {\mathcal{C}}_i$ the events $C_1,C_2,\dots$ are independent.