Probability that selection is random Suppose we have a base set $[1;n] \subset \mathbb{N}$ of size $n$. From this set $m$ numbers $S = (i_1,i_2,...,i_m)$ are picked. The same number may be picked multiple times. $m$ may be greater than $n$ or not.
It is conjectured that the selected numbers were chosen randomly, i.e. with evenly distributed probability and independently between draws.
How could I determine the probability that this is indeed the case?
The numbers are ordered, but if it simplifies things, this fact may be ignored.
 A: It depends on how strict that you want your test to be.  Probably the most straightforward test would be to use a chi-squared test.  You have an expected number of times each element of your base set should appear (m/n), and an observed number.  From this you can compute your chi-squared statistic and run a chi-squared test to see if the distribution is consistent with being uniform.
You could get fancier if you want, though.  The chi-squared test won't necessarily indicate whether or not each draw is independent from the last, for instance.  If m >> n and S_(i+1) = S_i + 1 mod n, then the resulting dataset will be consistent with a uniform distribution according to the chi-squared test, even though it's completely deterministic.  One of the more comprehensive tests of pseudo-random number generators is the Diehard test suite.  (Testing a pRNG is pretty much your situation since drawing from a pRNG should be uniform and independent.)  The Diehard test suite looks for all sorts of things like the distribution of lengths of ascending and descending runs, what happens if you use the draws to play craps (the dice game), and so on.
