Kolmogorov axioms: the foundation of probability
In an axiomatic approach, probability is not defined by “equally likely outcomes” or by intuition.
Instead, we start with a sample space \(\Omega\) (the set of all possible outcomes), a collection of events \(\mathcal{F}\)
(subsets of \(\Omega\)), and a function \(P:\mathcal{F}\to\mathbb{R}\) that assigns a number to each event.
Kolmogorov’s axioms specify the minimum rules that \(P\) must satisfy to behave like a probability measure.
These axioms are powerful because they let us prove the standard rules of probability (complements, unions, conditional probability, etc.)
without assuming any particular model.
Axiom 1: Non-negativity
For every event \(E\in\mathcal{F}\), we require \(P(E)\ge 0\).
This matches the interpretation of probability as a “size” or “mass” of an event.
If any event has a negative probability, the assignment cannot represent a probability measure.
In computations, small negative values sometimes appear because of rounding, so practical verifiers often allow a tiny tolerance.
Axiom 2: Normalization
The sample space itself must have probability one: \(P(\Omega)=1\).
This encodes the idea that “something in \(\Omega\) happens” with certainty.
In finite models where \(\Omega\) is partitioned into mutually exclusive outcomes \(E_1,\dots,E_n\),
normalization typically appears as a sum rule:
\[
\sum_{i=1}^n P(E_i)=1.
\]
Partition mode in this tool is designed for that common situation: you enter probabilities that represent a full finite sample space.
Axiom 3: Additivity for disjoint events
If two events \(A\) and \(B\) cannot happen together (they are disjoint: \(A\cap B=\varnothing\)),
then the probability of “\(A\) or \(B\)” must add:
\[
P(A\cup B)=P(A)+P(B).
\]
This idea extends to more than two disjoint events. In a finite setting, if \(A_1,\dots,A_k\) are disjoint,
then \(P(\bigcup_{i=1}^k A_i)=\sum_{i=1}^k P(A_i)\).
In general mode, the verifier lets you specify which unions you claim are disjoint so it can test this equality.
If you only provide the individual event probabilities but do not provide \(P(\text{union})\),
the tool can compute the sum but cannot fully confirm equality without additional information.
University-level extension: \(\sigma\)-additivity (measure theory tease)
The full Kolmogorov axiom is stronger: for any countable collection of pairwise disjoint events \((A_i)_{i\ge 1}\),
\[
P\!\left(\bigcup_{i=1}^{\infty} A_i\right)=\sum_{i=1}^{\infty} P(A_i).
\]
This is called \(\sigma\)-additivity and connects probability directly to measure theory.
Many deep results in probability (limits of events, convergence theorems, random variables as measurable functions)
rely on this countable structure.
How to use this verifier
Use Finite partition when your list represents an exhaustive set of mutually exclusive outcomes.
The tool checks non-negativity and whether the total matches \(P(\Omega)=1\).
Use General events when you have named events and want to verify additivity for specific disjoint unions you provide.
The animation helps you see how probabilities “accumulate” toward the total while the axiom checklist marks Pass/Fail
(or “Needs info” when more data is required to test a statement).
