The purpose of the paper is to illustrate a “give and take” type of interaction between the frequentist theory of probability and statistics, on the one hand, and research in science, on the other. Invariably, research in science involves some observationsxand the unknown “true” state of nature σ. Quite often, replication of the experiments reveals considerable variation inx, indicating that no mathematical treatment of the problem is possible without the assumption thatxis a sample value of a random variableX, treated in terms of frequentist theory of probability. As to the true state of nature, σ, situations vary. Indeed, there are cases where it appears natural to consider that σ is selected at random out of a certain known set Σ, with either known or unknown probability law. Section 2 lists three typical examples in which the frequentist theory of probability can “give” something to science. In each example, however, the assumption of the randomness of σ appears extraneous. Section 3 describes another example of research in science, again with a non-random Σ, which happened to “give” something to the frequentist theory of statistics. The motivation to reduce the frequency of erroneous conclusions in the general circumstances of the biological study had led to considering the possibility that a certain derivative may fail to exist. This created a problem of testing (Section 4) which does not appear to have been considered. A solution of the problem, in the form of optimal symmetric (Cα) tests, is given in Section 5.