Frequentism is the philosophy that probabilities are statistical in the sense that they give the limiting frequency ratios of outcomes as the number of trials is large enough. For tiny probabilities like exponentially small probabilities, would this require exponentially many trials?
I know, I know, you are thinking if the probability of an outcome is exponentially small, then we won't expect it to happen in any trial if the number of trials is some reasonable practical number. This overlooks the situation where say the possible outcomes are some long strings of characters taken from some alphabet in the computer science sense, and the probability distribution is such that the probability for any particular string is exponentially small. In such a case, if someone were to specify some specific string in advance, and the number of trials is some realistic number, no one would expect to find that string in any trial outcome. On the other hand, if you were to look at the trial outcomes, they would all correspond to strings which if chosen a priori, would be considered practically improbable. In practice, what experimenters do in such a situation is conduct statistical randomness tests upon the strings obtained, but does this take us out of the realm of frequentism?
Can frequentism be saved in such a case?