DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

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The uniform probability space on a nonempty finite set

Definition

Let $Ω$ be a nonempty finite set. Its uniform probability space assigns every $ω \in Ω$ the weight $1 / ∣ Ω ∣$ . Thus every event $A \subseteq Ω$ has $P (A) = \frac{∣ A ∣}{∣ Ω ∣} .$ Nonemptiness is required so that the denominator is positive.

Depends on

Used by

Events with the same probability need not be independent Counterexample
Expectation equal to 1 does not force a nonnegative integer-valued variable to vanish somewhere Counterexample
For dependent variables, E[XY] need not equal E[X]E[Y] Counterexample
Markov's conclusion can fail without nonnegativity Counterexample
Uncorrelated finite random variables need not be independent Counterexample
A symmetric two-point distribution attains equality in Chebyshev's inequality Example
The union bound can be strict for overlapping events Example
Three pairwise-independent events that are not mutually independent Example
False: linearity of expectation requires independence False statement
Every nonempty finite set of n nonzero integers has a sum-free subset of size greater than n/3 Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 54 results over 22 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Matousek and J. Vondrak, The Probabilistic Method, Section 1.1 (standard reference, not scraped)
M. Bucic, Probabilistic Method, Example A.2 (standard reference, not scraped)