Uniform Distribution: Equal Probability Explained Simply

The uniform distribution is the simplest probability distribution: every outcome in a range is equally likely. A fair die is the discrete version, each face has probability 1/6. The continuous version describes things like the exact minute a bus arrives between 0 and 10 minutes, where any instant in that window is equally probable.

It matters as the foundation of randomness itself. Computer random-number generators produce uniform values first, and every other distribution is built from them. In exams it is the cleanest place to learn probability density, because the density is just a flat rectangle, so areas become simple length ratios.

Key Concepts

• Discrete uniform: X takes values 1, 2, …, n, each with P(X = k) = 1/n. Mean = (n+1)/2, variance = (n² - 1)/12.
• Continuous uniform: X ~ Uniform(a, b). The density is f(x) = 1/(b - a) for a ≤ x ≤ b, and 0 elsewhere. Mean = (a + b)/2, variance = (b - a)²/12.
• CDF: P(X ≤ x) = (x - a)/(b - a) for x between a and b.
• Probability as length: P(c ≤ X ≤ d) = (d - c)/(b - a), the favourable length over the total length.
• No memory, no centre preference: uniform values do not cluster and do not favour any point; the histogram is flat.

Formulas

Continuous: f(x) = 1/(b - a), mean μ = (a + b)/2, variance σ² = (b - a)²/12. Discrete: P(X = k) = 1/n, mean (n + 1)/2, variance (n² - 1)/12. The only assumptions are that the range is finite and every value in it is equally likely. Note the variance grows with the square of the range width, so a wider uniform interval spreads values much more.

Worked Example

A metro train arrives at a random time between 0 and 10 minutes from now, uniformly. (a) Probability you wait more than 6 minutes: favourable length = 10 - 6 = 4, total = 10, so P = 4/10 = 0.4. (b) Mean waiting time = (0 + 10)/2 = 5 minutes. (c) Variance = (10 - 0)²/12 = 100/12 ≈ 8.333, so the standard deviation is √8.333 ≈ 2.887 minutes. For a die (discrete, n = 6): P(X = 4) = 1/6, mean = 7/2 = 3.5, variance = 35/12 ≈ 2.917.

Common Misconceptions

• Uniform is not "random-looking": people expect random numbers to spread out evenly, but small samples of uniform draws still show clusters and gaps. Only large samples flatten out.
• Continuous uniform has zero point probabilities: P(X = exactly 3.000) = 0 for a continuous variable. Only intervals have positive probability, unlike the discrete die where P(X = 4) = 1/6.
• The variance formula needs the square: variance is (b - a)²/12, not (b - a)/12. Forgetting the square is the most common arithmetic slip.

Real-World Applications

• Simulation: Monte Carlo methods start by drawing uniform random numbers, then transform them into any needed distribution.
• Lotteries and raffles: a fair draw from numbered tickets is a discrete uniform sample.
• Rounding error: the error when a measurement is rounded to the nearest unit is modelled as Uniform(-0.5, 0.5).

The uniform distribution is equal probability over a range: density 1/(b - a), centre (a + b)/2, and spread (b - a)²/12. Because probabilities reduce to length ratios, it is the clearest window into how density works, and the raw material from which all computer randomness is built.