The frequentist interpretation of probability states that the probability of a currently unrealized event occurring is the limit of its relative (usually temporal) frequency in a large number of historical "trials of the event". More prosaically, if I flip any coin a large number of times and find that the number of heads has approached 50%, then I should expect that my next flip will land on heads 50% of the time.
Do the "trials of the event" have to be precisely identical and if they're not, does that make real world (i.e. physical, non-mathematical) application of the frequentist interpretation rigorously invalid?
Obviously, there are many possible causes that prevent any temporal sequence of real world trials from being precisely identical. Foremost, I will be unable to exactly control the angular momentum I impart to the coin for every trial. In fact, I can't say this for sure, but I suspect that even "differential" changes in imparted angular momentum combined with the chaotic behavior of the coin-universe system will lead to "unpredictable" outcomes. Such behavior in fact is probably what we mean by "random" experiment vs. Laplacian determinism even w/o recourse to quantum mechanics. Another example of the impossibility of precisely identical trials is the effect of the gravitational pull of Jupiter since it has moved in space across two trials.
So, rigorously, why should I be able to say anything at all about the probability of the next event given that I haven't replicated the exact experiment.
Pragmatically, I realize that the frequentist approach "just works" for many real world applications (such as flipping a given coin). This is probably due to the multitude of effects offsetting each other, but I am interested in the narrow philosophical underpinnings of the interpretation.