A discrete uniform probability distribution is one in which all elementary events in the sample space have an equal opportunity of occurring. As a result, for a finite sample space of size *n*, the probability of an elementary event occurring is 1/*n*. Uniform distributions are very common for initial studies of probability. The histogram of this distribution will look rectangular in shape.

## Examples

One well-known example of a uniform probability distribution is found when rolling a standard die. If we assume that the die is fair, then each of the sides numbered one through six has an equal probability of being rolled. There are six possibilities, and so the probability that a two is rolled is 1/6. Likewise, the probability that a three is rolled is also 1/6.

Another common example is a fair coin. Each side of the coin, heads or tails, has an equal probability of landing up. Thus the probability of a head is 1/2, and the probability of a tail is also 1/2.

If we remove the assumption that the dice we are working with are fair, then the probability distribution is no longer uniform. A loaded die favors one number over the others, and so it would be more likely to show this number than the other five. If there is any question, repeated experiments would help us to determine if the dice we are using are really fair and if we can assume uniformity.

## Assumption of Uniform

Many times, for real-world scenarios, it is practical to assume that we are working with a uniform distribution, even though that may not actually be the case. We should exercise caution when doing this. Such an assumption should be verified by some empirical evidence, and we should clearly state that we are making an assumption of a uniform distribution.

For a prime example of this, consider birthdays. Studies have shown that birthdays are not spread uniformly throughout the year. Due to a variety of factors, some dates have more people born on them than others. However, the differences in popularity of birthdays are negligible enough that for most applications, such as the birthday problem, it is safe to assume that all birthdays (with the exception of leap day) are equally likely to occur.