A uniform distribution describes a situation where every allowed outcome is equally likely. It is one of the simplest probability models, but it is also one of the most useful because it gives a clear baseline for fairness and randomness. Dice, cards, spinners, random number generators, and measurement errors are often modeled with uniform distributions when no outcome is favored.
Its graph is flat because probability is spread evenly across the possible outcomes.
There are two main types: discrete uniform distributions and continuous uniform distributions. A discrete uniform distribution has a countable list of outcomes, such as the numbers 1 through 6 on a fair die. A continuous uniform distribution covers every value in an interval, such as a spinner landing anywhere from 0 to 360 degrees.
In both cases, the mean is at the center of the possible values, while the variance measures how spread out the outcomes are around that center.
Key Facts
- Discrete uniform probability: P(X = x) = 1/n for each of n equally likely outcomes.
- Continuous uniform density on [a, b]: f(x) = 1/(b - a) for a <= x <= b.
- Continuous uniform probability over an interval [c, d]: P(c <= X <= d) = (d - c)/(b - a).
- Discrete uniform mean for values 1 to n: μ = (n + 1)/2.
- Continuous uniform mean on [a, b]: μ = (a + b)/2.
- Continuous uniform variance on [a, b]: σ^2 = (b - a)^2/12.
Vocabulary
- Uniform distribution
- A probability distribution in which all allowed outcomes are equally likely.
- Discrete uniform distribution
- A uniform distribution with a finite or countable set of separate possible outcomes.
- Continuous uniform distribution
- A uniform distribution in which any value in a continuous interval can occur.
- Probability density
- A function that describes how probability is spread over a continuous range of values.
- Variance
- A measure of how far values in a distribution tend to spread away from the mean.
Common Mistakes to Avoid
- Treating density as probability is wrong because a continuous value has probability 0 at any single exact point. Probability comes from area over an interval.
- Forgetting that the total probability must be 1 is wrong because all probabilities across the full sample space must add or integrate to 1.
- Using the continuous variance formula for a discrete die is wrong because σ^2 = (b - a)^2/12 applies to a continuous interval, not separate outcomes like 1, 2, 3, 4, 5, and 6.
- Assuming every flat-looking graph is uniform is wrong because the distribution must be flat over its whole allowed range and zero outside that range.
Practice Questions
- 1 A fair 8-sided die is numbered 1 through 8. What is P(X = 5), and what is the mean value of one roll?
- 2 A spinner is uniformly distributed from 0 degrees to 360 degrees. What is the probability that it lands between 90 degrees and 150 degrees?
- 3 A computer generates a random decimal uniformly between 0 and 1. Explain why the probability of getting exactly 0.5 is not the same as the probability of getting a value between 0.4 and 0.6.