Uniform distribution, in full
The discrete uniform distribution over n outcomes gives each of them probability 1/n. That is the formal content of the word fair. A fair d6 is uniform over its six faces at 1/6 each, a fair d20 is uniform over twenty faces at 1/20, and a properly run 6-from-49 draw is uniform over its 13,983,816 combinations. The mean of a uniform distribution on 1 to n is (n+1)/2, giving 3.5 for a d6 and 10.5 for a d20.
Uniformity in a real draw is an engineering claim rather than a mathematical one. Ball sets are weighed and measured, machines are tested, draws are filmed and results audited, all to stop any single ball being favoured. Certified software draws chase the same target by a different route. No physical process is perfectly uniform, so operators test for drift and retire equipment that fails rather than assert perfection.
There is a continuous version too, uniform on an interval, where every sub-interval of equal width is equally likely. Draw games and dice are discrete, so the finite version is the one that matters here. A process can be perfectly uniform over its outcomes and still throw lopsided-looking summaries once outcomes are grouped: two d6 are uniform over 36 ordered pairs, and their totals are nowhere near uniform over 2 to 12.
- Probability
- Discrete uniform distribution, Equiprobable distribution
A worked example
A d20 is uniform if each of the twenty faces has probability 1/20 = 0.05. Roll it 200 times and the expected count per face is 10, but counts spread from 4 to 17 across the twenty faces would be ordinary variation, not evidence of a loaded die. Uniformity describes the probabilities, not a promise that any finite run of rolls comes out flat.
See also
Where it comes up on this site
Uniform distribution: frequently asked
That is the design target, taken over combinations rather than individual balls: each of the 13,983,816 sets in a 6-from-49 game at 1/13,983,816. Physical uniformity is maintained rather than assumed, which is why ball sets get weighed and machines get tested and pulled from service when they drift.
Because uniform is a statement about probabilities, not about counts. Twenty rolls of a d6 will almost never split three per face.