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.

First axiom
The probability of an event is a non-negative real number:
 * $$P(E)\geq 0 \qquad \forall E\subseteq \Omega$$

where $$\Omega$$ is the sample 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 &sigma;-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 &sigma;-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 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.