Probability axioms

Kolmogorov axioms:

An axiomatic approach is taken by Kolmogorov to define probability.

Let S denote as event space with a probability measure P defined over it, such that probability of any event E ∈ S is given by P(E). Then, the probability measure obeys the following axioms:

• Axiom 1: The probability of an event is a real number greater than or equal to 0.
  P(E) ≥ 0, the probability of an event is a non-negative real number.
  
• Axiom 2: The probability that at least one of all the possible outcomes of a process (such as rolling a die) will occur is 1.
  P(S) = 1, the probability that at least one of the elementary events in the entire sample space will occur is 1.
  
• Axiom 3: If two events A and B are mutually exclusive, then the probability of either A or B occurring is the probability of A occurring plus the probability of B occurring.
  If {E1, E2, … Ej, …} is a sequence of mutually exclusive events such that Ei ∩ Ej = ∅ for all i, j, then P(E1 ∪ E2 ∪ .. ∪Ej ∪ ..) = P(E1)+P(E2)+ … + P(Ej ) + …, the probability of an event occurring is the sum of the probabilities.