What is a probability function, and what properties must it satisfy?

Probability quantifies the degree of uncertainty in a random experiment, and a probability function maps elements in the sample space, or the mutually exclusive set of all possible outcomes of an experiment, to the interval [0,1]. In order for it to be a valid probability function, it must satisfy three criteria, which are commonly referred to as the Kolmogorov Axioms. 

  • The probability of an event A occurring is between 0 and 1, or 0 <= P(A) <= 1
  • The probability of some valid result occurring is 1, denoted P(S)=1, where S refers to the set of all possible outcomes
  • For any pair of mutually exclusive events, the probability of either of them occurring can be found by taking the sum of the probability of either occurring. If A and B are two outcomes of the experiment,

    P(A U B) = P(A) + P(B)