In Kolmogorov's probability theory, the probability P of some event E, denoted
Contents
- First axiom
- Second axiom
- Third axiom
- Consequences
- The probability of the empty set
- Monotonicity
- The numeric bound
- Proofs
- Further consequences
- Simple example coin toss
- References
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.
First axiom
The probability of an event is a non-negative real number:
where
Second axiom
This is the assumption of unit measure: that the probability that at least one of the elementary events in the entire sample space will occur is 1.
Third axiom
This is the assumption of σ-additivity:
Any countable sequence of disjoint sets (synonymous with mutually exclusive events)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.
Consequences
From the Kolmogorov axioms, one can deduce other useful rules for calculating probabilities.
The probability of the empty set
In some cases,
Monotonicity
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
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
Since the left-hand side of this equation is a series of non-negative numbers, and since it converges to
If
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:
An extension of the addition law to any number of sets is the inclusion–exclusion principle.
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.