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Probability axioms
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In probability theory, the probability P of some event E, denoted , is defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov.
These assumptions can be summarised as: Let (O, F, P) be a measure space with P(O)=1. Then (O, F, P) is a probability space, with sample space O, event space F and probability measure P.
probability of an event is a non-negative real number:
where is the event space.
is the assumption of unit measure: that the probability that some elementary event in the entire sample space will occur is 1.
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Encyclopedia
In probability theory, the probability P of some event E, denoted , is defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov.
These assumptions can be summarised as: Let (O, F, P) be a measure space with P(O)=1. Then (O, F, P) is a probability space, with sample space O, event space F and probability measure P.
First axiom
The probability of an event is a non-negative real number:
where is the event space.
Second axiom
This is the assumption of unit measure: that the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space.
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This is often overlooked in some mistaken probability calculations; if you cannot precisely define the whole sample space, then the probability of any subset cannot be defined either.
Third axiom
This is the assumption of σ-additivity:
- Any countable sequence of pairwise disjoint events satisfies
Some authors consider merely finitely additive probability spaces, in which case one just needs an algebra of sets, rather than a σ-algebra.
Consequences
From the Kolmogorov axioms, one can deduce other useful rules for calculating probabilities:
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This is called the addition law of probability, or the sum rule.
That is, the probability that A or B will happen is the sum of the
probabilities that A will happen and that B will happen, minus the
probability that both A and B will happen. This can be extended to the inclusion-exclusion principle.
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That is, the probability that any event will not happen is 1 minus the probability that it will.
See also
Further reading
- Von Plato, Jan, 2005, "Grundbegriffe der Wahrscheinlichtkeitsrechnung" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 960-69. (in English)
External links
- Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A.N. Kolmogorov. A.N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A.N. Kolmogorov.
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