Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Finite probability spaces, outcome weights, events, and event probabilities
Definition
A finite probability space is a pair consisting of a finite set and a function such that for every and The elements of are outcomes. Every subset is an event, and its probability is
An outcome of weight zero remains an outcome. Thus an event can be nonempty and still have probability zero.
Depends on
Used by
- Conditional probability ℙ(A∣ B) for ℙ(B)>0 Definition
- Independent events, pairwise independence, and mutual independence of a finite family Definition
- Real random variables on finite probability spaces and their finite distributions Definition
- The finite product of finite probability spaces Definition
- The uniform probability space on a nonempty finite set Definition
- A loaded die as a nonuniform finite probability space Example
- A two-valued random variable attains equality in Markov's inequality Example
- Most finite strings are incompressible Lemma
- Normalization, nonnegativity, monotonicity, complements, and differences in a finite probability space Lemma
- Agreement with the published finite probability-space definition Remark
- Finite probability spaces are exactly finite full-power-set probability spaces Theorem
- Probability is additive on every finite pairwise-disjoint family of events Theorem
Dependency tree · two levels
30 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 1.2 (standard reference, not scraped)
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Sections 1.3.2-1.3.3 (standard reference, not scraped)