Probability provides a formal mathematical framework for reasoning about processes whose individual outcomes cannot be predicted with certainty.
The modern foundation of probability rests on three axioms formulated by mathematician Andrey Kolmogorov in the nineteen thirties.
First, the probability of any event must be a non-negative real number greater than or equal to zero.
Second, the probability of the entire sample space occurring must equal exactly one.
Third, for mutually exclusive events, the probability of either event occurring equals the sum of their individual probabilities.
The sample space encompasses the exhaustive set of every conceivable elementary outcome that an experiment can produce.
An event is defined formally as any measurable subset of that sample space.
When all elementary outcomes share identical likelihood, calculating probability reduces to counting favourable outcomes over total outcomes.
For example, in a standard deck of fifty-two playing cards, drawing a spade has thirteen favourable outcomes out of fifty-two, equalling one quarter.
In continuous settings, individual points have a probability of zero, requiring probability to be calculated over intervals instead.
Two distinct philosophical interpretations dominate modern practice: the frequentist view and the Bayesian view.
Frequentists define probability as the limiting relative frequency of an event over an infinite sequence of identical trials.
Bayesians interpret probability as a subjective degree of belief based on current knowledge and available information.
Both schools of thought rely on the exact same underlying mathematical axioms and algebraic rules.
A common error is assuming that impossible events and events with zero probability are identical concepts in continuous spaces.