Twin Twister 2 - An English axiom of choice

When we are given a statement like "At least K out of N objects are <property 1>" (1), we might say something like "Then the other ones must be <property 2>" (2). When we say "other ones", we are selecting, presumably, a set "N\K" with size N-K. The problem is, this implies that there is some choice of K to begin with, when really all we gave was some statistic regarding the whole N-set. 

However, there are times when (2) is true no matter what our choice of the K-set is, as long as the K-set satisfies <property 1>, e.g. in the Twin Twister problem. 

But what are the times when it does matter? 

And in the Twin Twister problem, we have probabilities, with the statement "Then the other ones must be <property 2> with probability P." How does this change the problem? 

Is the probability computed over a set of outcomes spanning the different possible choices of K? If yes, how should those different choices be weighted? If not, is there something contextual that provides a canonical choice of K (e.g. say the vet knows something)?  Given that we find a way to choose K, is there some identifying factor that allows us to compute P over an outcome space that holds K fixed (e.g. identifying code on the lambs, and the outcomes all have this invariant -- thus turning the probability to 1/2)?

A general problem is this: A non-probabilistic, 0th order sentence S has a singular interpretation in a single context, but in the scope of hypothetical realities and possibilities, we might run into a situation where there is no "canonical" interpretation over them all -- i.e. an "atom" or "object" in a sentence may not necessarily correspond to a singularly identifiable "object" in all of the possibilities. There may even be multiple levels to this: e.g. there are multiple choices of K, and multiple possibilities of gender assignment. Unless the problem is conditioned so that we restrict ourselves to the outcome space where the lambs are identified based on genetic code, we run into a "choice function" scenario.