The Discrete Uniform Distribution

A discrete random variable $X$ follows a discrete uniform distribution on the finite set $\{x_1, x_2, \ldots, x_n\}$ if each of the $n$ values is equally likely. Its probability mass function is

$$P(X = x_i) = \dfrac{1}{n} \quad \text{for } i = 1, 2, \ldots, n.$$

The most common case is $X \sim \text{Uniform}\{1, 2, \ldots, n\},$ where

$$P(X = k) = \dfrac{1}{n} \quad \text{for } k = 1, 2, \ldots, n.$$

For example, rolling a fair six-sided die gives $X \sim \text{Uniform}\{1, 2, 3, 4, 5, 6\},$ so

$$P(X = 4) = \dfrac{1}{6}.$$

To compute the probability of an event $A,$ count the number of values in $A$ and divide by $n{:}$

$$P(X \in A) = \dfrac{|A|}{n}.$$

For the fair die, $P(X \geq 5) = \dfrac{|\{5,6\}|}{6} = \dfrac{2}{6} = \dfrac{1}{3}.$