Axiomatic definition of probability

Probability is a measure of uncertainty. Once a random experiment is defined, we call probability of the event A the real number $P\left(A\right)\in [0,1]$. The function $P(·):\Omega \to [0,1]$ is called probability measure or probability distribution and must satisfy the following three axioms:

• $P\left(A\right)⩾0;$ the probability is always a non-negative value.
• $P\left(\Omega \right)=1;$ the probability of the certain event is one (the maximum value).
• $\forall A,B/A\cap B=\varnothing \Rightarrow P(A\cup B)=P\left(A\right)+P\left(B\right);$ if two events are complementary, the probability of the union event equals the sum of probabilities.

The axiomatic definition of probability allows us to construct a robust theory of probability. To better understand the concept of probability we rely on the `Frequentist definition’:

• Frequentist definition of probability: The probability P(A) of an event A is the limit:
$P(A)=\underset{N\to \infty }{lim}\frac{N_A}{N}$
• Where N is the number of observations and N_A is the number of times that event A occurred.