Lotus BlogTheory - [T7]
Probability axioms
The Kolmogorov axioms are the foundations of probability theory introduced by Andrey Kolmogorov in 1933. The assumptions as to setting up the axioms can be summarised as follows: Let $(\Omega, F, P)$ be a measure space with $P(E)$ being the probability of some event $E$, and $P(\Omega) = 1$. Then $(\Omega, F, P)$ is a probability space, with sample space $\Omega$, event space $F$ and probability measure $P$.
First axiom
The probability of an event is a non-negative real number:
$$ P(E) \in \R, P(E) > 0, \forall E \in F $$
Where F is the event space. It follows that $P(E)$ is always finite, in contrast with more general measure theory. Theories which assign negative probability relax the first axiom.
In the same way relative frequency is also always a strictly positive value since it represents the ratio of an event occurring to the total number of events measured.
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
$$ P(\Omega) = 1 $$
In the same way when measuring the occurrences of certain events, the sum of all the relative frequencies computed for each event measured will add up to 1.
Third axiom
This is the assumption of $\sigma$-additivity: Any countable sequence of disjoint sets $E1,E2,\dots$ satisfies:
$$ P\Biggl( \bigcup_{i=1}^{\infty} E_i \Biggl) = \sum_{i=1}^{\infty} P(E_i) $$
As for relative frequency, when having two or more events, which cannot happen simultaneously, the sum of their relative frequencies is equal to the frequency of either event to happen. e.g.
$$ A \bigcap B = \empty \\ f(A) + f(B) = f(A|B) $$