In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein each of some finite whole number n of outcome values are equally likely to be observed.

Intuitively, a discrete uniform distribution is "a known, finite number of outcomes all equally likely to happen."

A simple example of the discrete uniform distribution comes from throwing a fair six-sided die.

Although it is common to consider discrete uniform distributions over a contiguous range of integers, such as in this six-sided die example, one can define discrete uniform distributions over any finite set.

For instance, the six-sided die could have abstract symbols rather than numbers on each of its faces.

Less simply, a random permutation is a permutation generated uniformly randomly from the permutations of a given set and a uniform spanning tree of a graph is a spanning tree selected with uniform probabilities from the full set of spanning trees of the graph.

However, in the common case that its possible outcome values are the integers in an interval

from a sample of k observations is commonly known as the German tank problem, following the practical application of this maximum estimation problem, during World War II, by Allied forces seeking to estimate German tank production.

A uniformly minimum variance unbiased (UMVU) estimator for the distribution's maximum in terms of m, the sample maximum, and k, the sample size, is

If samples from a discrete uniform distribution are not numbered in order but are recognizable or markable, one can instead estimate population size via a mark and recapture method.

Uniform discrete distributions over bounded integer ranges do not constitute an exponential family of distributions because their support varies with their parameters.

For families of distributions in which their supports do not depend on their parameters, the Pitman–Koopman–Darmois theorem states that only exponential families have sufficient statistics of dimensions that are bounded as sample size increases.

The uniform distribution is thus a simple example showing the necessity of the conditions for this theorem.