Probability axioms

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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.

Contents

[edit] First axiom

The probability of an event is a non-negative real number:

P(E)\geq 0 \qquad \forall E\in \Omega.

[edit] Second axiom

The probability that some elementary event in the entire sample set will occur is 1. More specifically, there are no elementary events outside the sample set.

P(\Omega) = 1.\,

This is often overlooked in some mistaken probability calculations; if you cannot precisely define the whole sample set, then the probability of any subset cannot be defined either.

[edit] Third axiom

Any countable sequence of pairwise disjoint events E1,E2,... satisfies P(E_1 \cup E_2 \cup \cdots) = \sum_i P(E_i).

This is called σ-additivity. Some authors consider merely finitely-additive probability spaces, in which case one just needs an algebra of sets, rather than a σ-algebra.

[edit] 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 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.

P(\Omega\setminus E) = 1 - P(E)

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

[edit] External links

  • 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.
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