the discrete uniform distribution

The discrete uniform distribution is the formal name for 'every outcome is equally likely', applied to a finite list of possibilities. A fair die has six faces, each with probability 1/6. A fair coin has two sides, each 1/2. Drawing a random card from a well-shuffled deck, picking a winning raffle ticket, generating a random integer between 1 and 100 — all are discrete uniform. It is the distribution of pure, unweighted chance over a finite set.

If there are n possible values, the pmf simply assigns each one probability 1/n; nothing is favoured. For the standard case where the values are the integers 1 through n, the formulas are tidy: the mean is E[X] = (n + 1)/2, the exact middle of the range, and the variance is Var(X) = (n^2 - 1)/12. You can read both off intuitively — the average of equally spaced values is their midpoint, and the variance grows like the square of the range, just as spread should. Because every outcome carries the same weight, computing any probability reduces to counting: P(event) = (number of favourable values)/n, which is exactly the classical definition of probability.

Its importance is partly conceptual and partly practical. Conceptually it is the baseline of symmetry — the model you assume when you have no reason to prefer one outcome over another, the foundation under all of classical combinatorial probability. Practically it is the engine of simulation: a computer's random number generator produces (approximately) uniform values, and every other distribution is then manufactured from these by transformation. The honest caveat is that real-world equiprobability is an assumption, not a fact — a slightly bent die or a biased generator quietly breaks it.