The three Kolmogorov axioms underpin probability theory. Before we explore the axioms, some common probability language will be introduced.
An experiment is a process of observation where the output cannot be predicted with certainty due to random effects. Example: rolling a dice.
A trial is a single occurrence of an experiment. Multiple trials of an experiment can form a new experiment. Example: an experiment consists of rolling a dice twice and the trial is one instance of the twice-rolled dice experiment.
An outcome is an observed output of a trial. Examples: rolling a 3 or rolling a 1 and 5 in the twice-rolled dice experiment.
The sample space Ω is the set of all possible outcomes of an experiment. Examples: each side of the dice or ordered pairs of twice-rolled dice experiments.
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An event A is a subset of outcomes in the sample space Ω. Examples: rolling a side less than 4, rolling an even number or rolling a 2 and then a 3.
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The operations of set theory apply to events. The union of events is the set of outcomes in A, B or both. The intersection of events is the set of outcomes in both A and B.
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The event consisting of no outcomes is called the null event. If events A and B have no outcomes in common then A and B are disjoint events (mutually exclusive).
A Probability measure P is a function that assigns a real number to each measurable event. A probability measure must follow the axioms of probability.
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Now we will explore the three axioms of probability.
First axiom: non-negative, real number
The probability of an event is a non-negative real number.
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This axiom means that the smallest probability of an event is zero. It does not specify an upper bound, however a probability theorem does.
Second axiom: unitarity
The probability that at least one outcome in the sample space will occur is 1.
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This axiom means that it is certain that an outcome will occur from observing an experiment.
Third axiom: countable additivity
If there is an infinite set of disjoint events in a sample space Ω then the probability of the union of events is equal to the sum of probabilities of all events.
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This axiom forms a relationship between a set of disjoint events in a sample space and the individual probabilities of each event. A probability theorem shows how a finite set of disjoint events can be represented as an infinite set too.
The axioms of probability can subsequently be used to derive the theorems of probability.