Editing Independence (probability theory) (section)

====More than two events====
A finite set of events <math>\{ A_i \} _{i=1}^{n}</math> is [[Pairwise independence|pairwise independent]] if every pair of events is independent<ref name ="Feller">{{cite book | last = Feller | first = W | year = 1971 | title = An Introduction to Probability Theory and Its Applications | publisher = [[John Wiley & Sons|Wiley]] | chapter = Stochastic Independence}}</ref>&mdash;that is, if and only if for all distinct pairs of indices <math>m,k</math>,

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>\mathrm{P}(A_m \cap A_k) = \mathrm{P}(A_m)\mathrm{P}(A_k)</math>|{{EquationRef|Eq.2}}}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

A finite set of events is '''mutually independent''' if every event is independent of any intersection of the other events<ref name="Feller" /><ref name=Gallager/>{{rp|p. 11}}&mdash;that is, if and only if for every <math>k \leq n</math> and for every k indices <math>1\le i_1 < \dots < i_k \le n</math>,

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>\mathrm{P}\left(\bigcap_{j=1}^k A_{i_j} \right)=\prod_{j=1}^k \mathrm{P}(A_{i_j} )</math>|{{EquationRef|Eq.3}}}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

This is called the ''multiplication rule'' for independent events. It is [[#Triple-independence but no pairwise-independence|not a single condition]] involving only the product of all the probabilities of all single events; it must hold true for all subsets of events.

For more than two events, a mutually independent set of events is (by definition) pairwise independent; but the converse is [[#Pairwise and mutual independence|not necessarily true]].<ref name=Florescu/>{{rp|p. 30}}