You may want to look at D. Ringe's On Calculating the Factor of Chance in Language Comparison, which lays out some of the problems. I believe that uncontrolled variables are the greatest impediment to subjecting word-relatedness questions to valid statistical testing. Moreover, the idea that one could ever compute a p-value that a given word of a modern language is the result of ... (what?) from a hypothesized earlier word is a conceptual abuse of what a p-value is. It is mathematically possible to cook data so that a program will return a p-value: the question is, what reality does the number represent?
As an example of an uncontrolled variable, we might conjecture that bandanna derives from the same PIE root as bind, and could enter both words in a PIE-to-English table of pairs that would also contain 'father; hound; wheel' and corresponding PIE roots. In doing this, we will have failed to control the fact that 'bind, father, hound, wheel' are words transmitted Genetically through (West) Germanic into modern English, but 'bandanna' is recently borrowed from Hindi. Uncertainty is created because etymological claims usually mean "following this transmission path", yet "bandanna" does not follow that path (instead, the path is via Indic then recently by contact between Indic and English).
Another factor that has to be controlled in constructing etymologies is meaning. When doing etymologies in Bantu (a relatively shallow time depth proto-language), the meanings of words from a given root are usually the same, so various descendants of *bón- mean "see". However, some of the words mean "find" or "acquire", thus you have to allow imprecision in semantics, to allow including the entire set of words derived through normal historical transmission from Bantu *bón-. We impose regularity-of-correspondence conditions on our etymological n-tuples, when it comes to the segments of the words – a Bantu word of the form cimb- does not go into the etymological database along with puk- just because the words mean "dig" (no set of historical changes turns puk- into cimb-). Rigorous semantic laws mapping classes of meanings of proto-languages to daughter languages are uncommon (to say the least – non-existent, as far as I know).
The claim that English "path" is a descendant of a PIE root that resulted in Latin pōns, Greek πόντος, OPrus. pintis, and O. Armenian hun would be excluded on the grounds of sound-change irregularity (the predicted regular outcome is something like "find"), but we can supplement the rules for admitting examples into the etymological database by positing (as we obviously must) that not all transmission paths are from parent to child – there are also borrowings. We might conjecture that English "path" is borrowed from some other language, such as Proto-Iranian. While this is very reasonable, it introduces an element of uncertainty. Subjectively, the uncertainty is not enough to render some alternative more plausible.
On the philosophical front, your question asks the fundamental epistemological question, "what is 'certainty'?; what is a theory of knowledge?". In history and philosophy of science, you are asking "what is 'probability'?" (there are different combinatoric, frequency and evidentiary senses). In statistics, where the really heavy lifting would be done, the problem is conceptualizing the question in a statistically valid way, taking care to understand what "the population" is, what the hypotheses would be, how independent variables are coded, what a "random sample" is, and what a p-value is. The concept of p-value simply is not applicable to the hypothesis that "hundred" derives through normal transmission from IE *ḱm̥tóm.
