How to Find Probabilities for a Uniform Distribution

A uniform distribution assigns equal probability to intervals of equal length within its range. For a continuous uniform random variable $X$ on $[a,b]$, the probability of landing in an interval is the portion of the total range covered by that interval.

This article gives the calculation for the continuous uniform distribution, the usual interpretation when a problem says that a quantity is “uniformly distributed.” If the possible outcomes are separate values, such as the integers on a die, use the discrete version described below.

Continuous uniform distribution

Write $X \sim \operatorname{Uniform}(a,b)$, where $a<b$. Its probability density is constant between $a$ and $b$:

The density is the same everywhere on the interval, so the probability of an event is the area under the density over that event. For an interval $[c,d]$, first find how much of it overlaps $[a,b]$. The result is

When $[c,d]$ is fully inside $[a,b]$, this simplifies to

The numerator is the length of the desired interval; the denominator is the length of the full range. If the event falls entirely outside the range, its probability is zero. If it covers the entire range, its probability is one.

Worked example

Suppose a bus arrives at a uniformly distributed time between 0 and 20 minutes after its scheduled time. Let $X$ be the delay in minutes, so $X\sim\operatorname{Uniform}(0,20)$. What is the probability that the bus arrives between 7 and 12 minutes late?

The full range has length $20-0=20$ minutes. The requested interval has length $12-7=5$ minutes, and it lies completely inside the range. Therefore,

The probability is $0.25$, or 25%. For another event, such as arriving more than 15 minutes late, the favorable length is $20-15=5$ minutes, giving $5/20=0.25$ as well.

If instead you asked for the probability that the bus arrives between 18 and 25 minutes late, only the part from 18 to 20 is possible. The overlap has length 2, so the probability is $2/20=0.10$.

Use the cumulative distribution function

The cumulative distribution function gives the probability that $X$ is at most a specified value $x$:

You can use it to calculate interval probabilities by subtracting cumulative probabilities. For example, when $a\le c\le d\le b$,

This approach is especially useful for events that involve “less than,” “more than,” or several intervals. For example, $P(X>c)=1-F_X(c)$ when $c$ is within the range.

What if the outcomes are discrete?

For a discrete uniform distribution, count equally likely outcomes instead of measuring interval lengths. If $X$ can be any integer from $m$ through $n$, inclusive, there are $n-m+1$ possible values. The probability of an event is

For a fair six-sided die, the probability of rolling a 2, 3, or 4 is $3/6=1/2$. Do not use the continuous length formula for a die: the outcomes are separate integers, not points along a continuous interval.

Common mistakes

• Using the density as a probability. For $X\sim\operatorname{Uniform}(a,b)$, the density is $1/(b-a)$, but a density value is not the probability of a particular outcome. Probabilities come from area: density multiplied by interval length.
• Forgetting to restrict the event to the range. If the requested interval extends beyond $[a,b]$, count only its overlap with $[a,b]$.
• Including endpoints in a continuous calculation. For a continuous variable, $P(X=c)=0$ for any single value $c$. Thus $P(c<X<d)$, $P(c\le X\le d)$, and versions that include just one endpoint are equal.
• Mixing up continuous and discrete cases. A continuous uniform distribution uses interval lengths; a discrete one uses counts of possible outcomes.

Check that your final probability lies between 0 and 1. For a continuous uniform variable, it should also equal the favorable length divided by the total length when the event is an interval inside the distribution’s range.