Definition
The probability axioms are the three rules that any probability assignment must obey, formalised by Andrey Kolmogorov in 1933:
- Non-negativity — every event A has probability P(A) ≥ 0.
- Normalisation — the entire sample space has probability P(Ω) = 1.
- Countable additivity — for any countable collection of mutually exclusive events, the probability of their union equals the sum of their individual probabilities.
These three statements are minimal but sufficient: every theorem of classical probability theory follows from them.
Why it matters
How it works
In practice, you specify a probability model by listing the sample space Ω and assigning a probability to each elementary outcome (or, in the continuous case, a density). Provided the assignment is non-negative and the total is 1, you have a valid probability measure. Every event's probability is then computed by summing (or integrating) over its outcomes.
The axioms guarantee internal consistency: if you obey them, your probabilities will never contradict each other no matter how you combine events. They are the mathematical guarantee that probability calculations 'add up'.