Suppose we have a mathematical statement P(x) related to a real number x in (a,b) And we have a method to determine the correctness of P(x) for any x in (a,b) If we randomly select x for finite times, and suppose all P(x) is true for choosen x. What we can say about the correctness of P(x)in(a,b) in perspective of sampling? Does it have a concrete theoretical foundation of similar problem?
2026-02-26 04:15:27.1772079327
Checking a mathematical statement with sampling method?
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Not really, since you can not test all possible outcomes. Depending on the exact hypothesis on $P$, you can get both false-positive and false-negative answers. The "hypothesis testing" part of statistics has various tools that estimate, e.g. "the probability to reject a valid hypothesis" or "if my hypothesis is true, what is the probability of the observed outcome of my tests". An article on confidence intervals can be a good start.