DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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
Unions of overlapping independent events need not remain independent Counterexample
A symmetric two-point distribution attains equality in Chebyshev's inequality Example
Independent events need not be disjoint 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
Small total induced-copy expectation forces many homogeneous k-sets Lemma
Every nonempty finite set of n nonzero integers has a sum-free subset of size greater than n/3 Theorem

Dependency tree · two levels

18 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

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)