A probability assigns nonnegative, normalised, countably additive weights to events.
Key knowledge
A probability space consists of an outcome set Ω, a sigma-algebra of events and a probability P. Probabilities are nonnegative, P(Ω)=1, and the probability of a countable union of pairwise disjoint events equals the sum of their probabilities.
Values and equations
P(Ω)=1; P(⋃ᵢ Aᵢ)=Σᵢ P(Aᵢ) for pairwise disjoint Aᵢ
Ω
Outcome space
P
Probability measure
Aᵢ
Events in a finite or countable disjoint family
Conditions and scope
Events belong to the specified sigma-algebra, which contains Ω and is closed under complements and countable unions.
These axioms constrain a probability model; they do not determine empirical probabilities without additional information.
Sources · 2
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
Event collection, normalisation and countable additivity.
2026-10-10 · Checked the general axioms; uniform outcomes are not assumed.
Foundational review
2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.
Sources from other places
Agreement within this scope
A sigma-algebra and nonnegative, normalised, countably additive probability.
The mathematical axioms describe a model; empirical probabilities require further information.
A probability space consists of an outcome set Ω, a sigma-algebra of events and a probability P. Probabilities are nonnegative, P(Ω)=1, and the probability of a countable union of pairwise disjoint events equals the sum of their probabilities.
For a discrete real-valued random variable X with probabilities p(x), its finite expected value is Σx xp(x) when Σx |x|p(x) is finite. The expected value need not itself be a possible observed value.
For a finite discrete distribution with probabilities pᵢ, Shannon entropy is −Σᵢ pᵢ log₂ pᵢ, measured in bits. A zero-probability term contributes zero by its limiting value.
For a real random variable X, F_X(x)=P(X≤x) is its cumulative distribution function. Discrete laws use probability masses; absolutely continuous laws use a nonnegative density integrated over events.
A hash table maps keys to buckets and stores colliding keys in a chain. With load factor α=n/m and a suitably randomized universal hash family, lookup takes expected O(1+α); an adversarial collision pattern can instead require O(n).
A sigma-algebra contains the whole space and is closed under complements and countable unions. A measure assigns nonnegative extended values, gives the empty set zero and is countably additive on disjoint measurable sets; sigma-finiteness means a countable cover by finite-measure sets.
For finite distributions p,q, D(p‖q)=Σx p(x)log₂[p(x)/q(x)]≥0, with equality exactly when p=q. It is asymmetric; positive p at a zero q gives infinite divergence.
For a vector query with global L1 sensitivity Delta-1, adding independent Laplace noise of scale Delta-1/epsilon to each coordinate gives epsilon-DP under the chosen adjacency. Postprocessing without new private-data access preserves that guarantee. Adaptive composition adds epsilon values and delta values in the basic bound; repeated releases consume privacy budget even if each release looks noisy. Utility depends on sensitivity, dimensions and the task's tolerated error.
PAC learning bounds population risk with probability at least 1−delta from iid samples. In realizable binary classification it targets risk at most epsilon when the class contains a zero-error rule. Agnostic learning instead targets excess risk at most epsilon above the best rule in the class. Neither definition says the world is in the model or permits replacing iid samples with arbitrary dependent deployment data.
Confidence is probability over training-sample randomness.
A qubit pure state is a normalized vector in C² modulo global phase; a mixed state is a positive trace-one density operator. A POVM consists of positive effects summing to the identity, with outcome probability tr(rho E-m). A density matrix can represent many ensemble decompositions, so it does not identify a unique hidden classical preparation. Quantum measurements and noncommuting observables cannot be modeled by one arbitrary classical sample space without further structure.
Born probabilities are normalized classical outcome probabilities.
A discrete martingale M-n is adapted to increasing sigma-algebras F-n, is integrable at every time, and obeys E[M-(n+1)|F-n]=M-n. For independent integrable centered increments, the partial sums form a martingale; for any integrable terminal X, E[X|F-n] is another. A filtration encodes information rather than a physical force or a prediction of each sample path.
The filtration belongs to a fixed probability space.
If E₁,…,Eₖ partition the outcome space, total probability sums P(Eᵢ)P(B|Eᵢ) to obtain P(B). Bayes' theorem then gives P(Eᵢ|B)=P(Eᵢ)P(B|Eᵢ)/P(B). Reversing a conditional probability therefore requires the prior event probabilities.
The disjoint partition allows probability addition.