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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 Russian mathematician Andrey Kolmogorov, which are described below.
The probability of an event is a non-negative real number:
This is the assumption of 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. Quasiprobability distributions in general relax the third axiom.
From the Kolmogorov axioms, one can deduce other useful rules for calculating probabilities.
The probability of the empty set
In some cases, is not the only event with probability 0.
If A is a subset of, or equal to B, then the probability of A is less than, or equal to the probability of B.
The numeric bound
It immediately follows from the monotonicity property that
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 left-hand side of this equation is a series of non-negative numbers, and since 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 .
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.
An extension of the addition law to any number of sets is the principle of inclusion-exclusion.
Setting B to the complement Ac of A in the addition law gives
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 coin-toss, 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.
- Borel algebra
- Set theory
- Conditional probability
- 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) (Learn how and when to remove this template message)|
- Von Plato, Jan, 2005, "Grundbegriffe der Wahrscheinlichkeitsrechnung" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 960-69. (in English)
- Glenn Shafer; Vladimir Vovk, The origins and legacy of Kolmogorov’s Grundbegriffe (PDF)