= Kolmogorov Axioms =

The '''Kolmogorov axioms''' are the basis for probability theory.

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== Description ==

One component of a [[Analysis/ProbabilitySpace|probability space]] triplet ''(Ω, ''ℱ'', P)'' is the '''probability measure''' ''P''. This is a [[Analysis/Functions|map]] that satisfies three conditions called the '''Kolmogorov axioms''':
 1. For any event ''A ∈'' ℱ, ''P(A) ≥ 0''
 2. ''P(''ℱ'') = 1''
 3. additivity (see [[Analysis/MeasureSpace|measure spaces]] for details)

Various rules follow from these axioms.

 * For any event ''A'', ''P(A) = 1 - P(A)^C^''
 * ''P(Ø) = 0''
 * If two events are characterized by ''A ⊆ B'', i.e. event ''A'' implies event ''B'', then ''P(A) ≤ P(B)''
 * For any event ''A ∈'' ℱ, ''P(A) ≤ 1''
 * For any two events ''A'' and ''B'', ''P(A ⋃ B``) = P(A) + P(B``) - P(A ⋂ B``)''
 * For any three events ''A'', ''B'', and ''C''; ''P(A ⋃ B ⋃ C) = P(A) + P(B``) + P(C) - P(A ⋂ B``) - P(A ⋂ C) - P(B ⋂ B``) + P(A ⋂ B ⋂ C)''



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