Uniform Distribution (Discrete)

The discrete uniform distribution has an outcome space over a range of values, each of which happens with equal probability.

$$P(X = x) =\begin{cases}1/(b - a + 1), &x \in \{a, ..., b\}\\0, &x \notin \{a, ..., b\}\end{cases}$$

A real-world system this distribution can model is rolling a fair 6 sided die. Here, a = 1 and b = 6, meaning that the die roll can result in a value from 1 to 6, each with a 1/6 probability of occurring.

Question

Which of the following can be reasonably modeled by the discrete uniform distribution?

(a) A biased coin flip where the probability of heads is less than the probability of tails

(b) Drawing a queen from a standard deck of 52 cards

(c) A raffle where each ticket corresponds to a unique person

Answer

(c) is the correct answer because every person is represented by one ticket, and hence everyone has an equal chance of winning. (a) and (b) are both events that occur with probabilities that are not equal to the probabilities of the other events in that outcome space.