Probability axioms


Probability axioms

In probability theory, the probability "P" of some event "E", denoted P(E), is defined in such a way that "P" satisfies the Kolmogorov axioms, named after Andrey Kolmogorov.

These assumptions can be summarised as: Let (Ω, "F", "P") be a measure space with "P"(Ω)=1. Then (Ω, "F", "P") is a probability space, with sample space Ω, event space "F" and probability measure "P".

First axiom

The probability of an event is a non-negative real number::P(E)geq 0 qquad forall Ein F

where F 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.: P(Omega) = 1.,

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 E_1, E_2, ... satisfies P(E_1 cup E_2 cup cdots) = sum_i P(E_i). 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:

: P(A cup B) = P(A) + P(B) - P(A cap B)

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 theprobabilities that "A" will happen and that "B" will happen, minus theprobability that both "A" "and" "B" will happen. This can be extended to the inclusion-exclusion principle.

: P(Omegasetminus E) = 1 - P(E)

That is, the probability that any event will "not" happen is 1 minus the probability that it will.

See also

* Cox's theorem

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

* [http://www.kolmogorov.com/ The Legacy of Andrei Nikolaevich Kolmogorov] 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.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Probability theory — is the branch of mathematics concerned with analysis of random phenomena.[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non deterministic events or measured… …   Wikipedia

  • Probability interpretations — The word probability has been used in a variety of ways since it was first coined in relation to games of chance. Does probability measure the real, physical tendency of something to occur, or is it just a measure of how strongly one believes it… …   Wikipedia

  • Probability — is the likelihood or chance that something is the case or will happen. Probability theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the… …   Wikipedia

  • probability theory — Math., Statistics. the theory of analyzing and making statements concerning the probability of the occurrence of uncertain events. Cf. probability (def. 4). [1830 40] * * * Branch of mathematics that deals with analysis of random events.… …   Universalium

  • Probability space — This article is about mathematical term. For the novel, see Probability Space (novel). In probability theory, a probability space or a probability triple is a mathematical construct that models a real world process (or experiment ) consisting of… …   Wikipedia

  • Bayesian probability — Bayesian statistics Theory Bayesian probability Probability interpretations Bayes theorem Bayes rule · Bayes factor Bayesian inference Bayesian network Prior · Posterior · Likelihood …   Wikipedia

  • List of axioms — This is a list of axioms as that term is understood in mathematics, by Wikipedia page. In epistemology, the word axiom is understood differently; see axiom and self evidence. Individual axioms are almost always part of a larger axiomatic… …   Wikipedia

  • Glossary of probability and statistics — The following is a glossary of terms. It is not intended to be all inclusive. Concerned fields *Probability theory *Algebra of random variables (linear algebra) *Statistics *Measure theory *Estimation theory Glossary *Atomic event : another name… …   Wikipedia

  • List of probability topics — This is a list of probability topics, by Wikipedia page. It overlaps with the (alphabetical) list of statistical topics. There are also the list of probabilists and list of statisticians.General aspects*Probability *Randomness, Pseudorandomness,… …   Wikipedia

  • Outline of probability — Probability is the likelihood or chance that something is the case or will happen. Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the… …   Wikipedia