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| * For any event ''A ∈'' ℱ, ''P(A) ≥ 0'' * ''P(''ℱ'') = 1'' * additivity (see [[Analysis/MeasureSpace|measure spaces]] for details) |
1. For any event ''A ∈'' ℱ, ''P(A) ≥ 0'' 2. ''P(''ℱ'') = 1'' 3. additivity (see [[Analysis/MeasureSpace|measure spaces]] for details) |
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| Various properties follow from these axioms. | Various rules follow from these axioms. |
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| * | * 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)'' ---- CategoryRicottone |
Kolmogorov Axioms
The Kolmogorov axioms are the basis for probability theory.
Contents
Description
One component of a probability space triplet (Ω, ℱ, P) is the probability measure P. This is a map that satisfies three conditions called the Kolmogorov axioms:
For any event A ∈ ℱ, P(A) ≥ 0
P(ℱ) = 1
additivity (see 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)