Discrete uniform distribution

Definition

Let $S$ be a nonempty finite set with $n$ elements. The discrete uniform distribution on $S$ is the distribution with the probability mass function $$p(s) = \frac{1}{n} \quad \text{for } s \in S.$$ A random variable $X$ is uniform on $S$ if $P(X = s) = 1/n$ for each $s \in S$.

Remarks

Then $P(X \in A) = |A| / n$ for each $A \subseteq S$. This is the classical definition of probability. For $S = \{a, a + 1, \dots, b\}$ with integers $a \le b$ and $n = b - a + 1$, the mean is $(a + b)/2$ and the variance is $(n^2 - 1)/12$. The result of a fair die is uniform on $\{1, \dots, 6\}$, with mean $7/2$.

Explanations

L1 human

The discrete uniform distribution is "all outcomes equally likely". A fair die, a fair coin and a well-shuffled deck all give uniform distributions.