Probability axioms
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Probability theory 


In Kolmogorov's probability theory, the probability P of some event E, denoted , is usually defined such that P satisfies the Kolmogorov axioms, named after the famous Russian mathematician Andrey Kolmogorov, which are described below.
These assumptions can be summarised as follows: 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.
An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem.
Contents
Axioms
First axiom
The probability of an event is a nonnegative real number:
where is the event space. In particular, is always finite, in contrast with more general measure theory. Theories which assign negative probability relax the first axiom.
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.
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 σadditivity:
 Any countable sequence of disjoint sets (synonymous with mutually exclusive events) satisfies
Some authors consider merely finitely additive probability spaces, in which case one just needs an algebra of sets, rather than a σalgebra. Quasiprobability distributions in general relax the third axiom.
Consequences
From the Kolmogorov axioms, one can deduce other useful rules for calculating probabilities.
The probability of the empty set
Monotonicity
The numeric bound
It immediately follows from the monotonicity property that
Proofs
The proofs of these properties are both interesting and insightful. They illustrate the power of the third axiom, and its interaction with the remaining two axioms. When studying axiomatic probability theory, many deep consequences follow from merely these three axioms. In order to verify the monotonicity property, we set and , where for . It is easy to see that the sets are pairwise disjoint and . Hence, we obtain from the third axiom that
Since the lefthand side of this equation is a series of nonnegative numbers, and that it converges to which is finite, we obtain both and . The second part of the statement is seen by contradiction: if then the left hand side is not less than infinity
If then we obtain a contradiction, because the sum does not exceed which is finite. Thus, . We have shown as a byproduct of the proof of monotonicity that .
Further consequences
Another important property is:
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. The proof of this is as follows:
 (by Axiom 3)
now, .
Eliminating from both equations gives us the desired result.
This can be extended to the inclusionexclusion principle.
That is, the probability that any event will not happen (or the event's complement) is 1 minus the probability that it will.
Simple example: coin toss
Consider a single cointoss, and assume that the coin will either land heads (H) or tails (T) (but not both). No assumption is made as to whether the coin is fair.
We may define:
Kolmogorov's axioms imply that:
The probability of neither heads nor tails, is 0.
The probability of either heads or tails, is 1.
The sum of the probability of heads and the probability of tails, is 1.
See also
 Law of total probability
 Measure (mathematics)
 Borel Algebra
 σAlgebra
 Probability theory
 Set theory
 Conditional probability
 Quasiprobability
 Fully probabilistic design
This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (November 2010) 
Further reading
 Von Plato, Jan, 2005, "Grundbegriffe der Wahrscheinlichkeitsrechnung" in GrattanGuinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 96069. (in English)
 Glenn Shafer; Vladimir Vovk, The origins and legacy of Kolmogorov’s Grundbegriffe (PDF)
External links
 Kolmogorov`s probability calculus, Stanford Encyclopedia of Philosophy.
 Formal definition of probability in the Mizar system, and the list of theorems formally proved about it.