What is "...", here? If it's infinite then we're talking about something abstract and you need to be more precise. If we're talking about something finite the

What is "...", here? If it's infinite then we're talking about something abstract and you need to be more precise. If we're talking about something finite then you need the specify the finite-ness. Note that (1,2,1,2,...,1,2) is not a "distribution," as you seem to be saying. It's a sequence of values drawn from a distribution. For a fixed length, is it possible for the sequence (1,2,1,2,1,2,...,1,2) to be drawn from a uniformly random distribution that contains 1 and 2 as possible values? Yes. How likely is it? It depends on the number of possible values in the distribution and the length of the sequence. The more values besides 1 and 2 and the longer the sequence, the less likely that sequence will have been drawn from a uniformly random distribution. In fact, as the length of the sequence increases, the probability of any particular sequence approaches 0 but is always greater than 0. The crux of your confusion appears to be that you think "distinguishable" means "a sequence that couldn't possibly have been drawn from a particular distribution," which is not the case.