A man created a list of 406 words, place and tribe names that he transcribed into English (Roman alphabet) and placed them side-by-side with ancient Hebrew names (that were likewise transcribed) to support a theory. I hope to objectively determine the likelihood of chance involvement. How can I proceed? I’m not a mathematician or a linguist, but it seems that math, text data mining and linguistics could help clarify. Some transcriptions are total “matches.” Others can be seen to represent possible pronunciation changes over time while still seeming to suggest a connection to ancient Hebrew. Perhaps such “near matches” could be assigned a lesser value to still support subjecting the list to analysis.
First level check: Factually correctness
Do the words mean what they are claimed to mean? Are they well-attested and not disputed (The Hebrew Bible contains some hapaxes [words occurring only once] with uncertain meanings)? Check with a dictionary. Do it for both sides of the list.
Second level check: Internal etymologies, loan words and borrowed words
Loan words and borrowed words aren't suitable for direct language comparison, so you have to exclude them. Use an etymological dictionary to identify them. Do this for both sides of the list.
Also watch for transparent etymologies such as prefixed and suffixed words as well as compounds. Only word stems are suited for language comparison, not the derived word forms.
Third level check: Regular sound correspondences
Watch for regular sound correspondences in the cleaned up word lists. Regular is the key word here: Singular correspondences are very likely chance correspondences, but larger families of words with the same shared sound change are much more convincing. Sound changes are often conditioned by their environment, for a starter I suggest comparing single word-initial consonants, they tend to give rather clear signals.
Sanity Check: Established language families
Take a look at some established language family to see how regular correspondences look like. Also take into account the assignment of the given languages to established language families (Semitic and Afro-Asiatic for Acient Hebrew, and whatever it is for the unnamed language in comparison).
Even a layman should be able to see the impressive correspondences laid out in this wikipedia article on Indo-European vocabulary. Contrast this with data from a non-Indogermanic language like Turkish, Japanese, or Tamil for an unrelated language.
P.S. The article quoted in @Draconis' comment is also worth reading.
An out of the blue comparison of two lists of letter-collections (I won't call them words) might well be provided to prove that two writing systems are computationally interchangeable: step one is always to identify what kind of claim is actually being made. The word "connection" should set off an alarm, because it is very difficult to prove that two things have absolutely no connection of any sort. For example if I give you a list of words in Indo-European languages which mean "woman", and a list of words from Bantu languages with the meaning "woman" then yes there is a connection between the words, but it is not historical, genetic or social, it's semantic.
So the first thing you have to do is prove that the author has a specific claim in mind, for example "this list of word pairs in Latin and Hebrew proves that Latin derives from Hebrew". Or, "this list of words shows that the Romans were capable of borrowing words from other languages". Once we exclude vague claims like "there is some connection between these two languages" (due to cultural contact), we are generally left with claims about descent from a common ancestor.
Prior to jk's first-level check, check the objective evidence for the existence of the words and their representations. The internet contains a lot of made-up data, so how do you know that there is an actual ancient manuscript with some set of squiggles, and how do you know how to interpret the individual squiggles? That's easy with <ω> and possibly impossible with some other sign. This is at least a documented symbol of the Phaistos disc, though it cannot be assigned a letter other than arbitrarily.
Second: there is definitely something suspicious about transcriptions being exact matches. There are multiple conventions for transcribing languages not written in the Latin alphabet. You have to consult the original texts of the ?two ?language, and determine if they follow a consistent and accepted system of transliteration – or is the transcription fine-tuned in order to support the hypothesis? (I should point out that under conventional linguistic usage, you "transcribe" from a sound source such as a speaker or a recording, and you "transliterate" from one alphabet to another, using a written text. Transcriptions are phonetic claims, transliterations are sometimes mistaken for being phonetic claims).
Then of course word-meaning in a foreign language is not self-evident, so how do they know that the word means such-and-such. Is it plausible that similar-looking words are because of borrowing (the various words across languages for "horse" that come from Arabic faras).
In moving to the question of chance in comparative linguistics, I recommend D. Ringe, "On calculating the factor of chance in language comparison" (Trans. Am. Philos. Soc. 82, 1–110). The biggest problem is that at least as practiced, comparison of the type that you describe have massive numbers of subconsciously-controlled variables. There is a root in PIE with a form something like pet which is claimed to mean a range of things like "fly" or "rush". It shows up in English in words like feather; petulant; pterodactyl; hippopotamus. Given other evidence, this semantically diverse collection is not the nonsense that it seems to be. If you require strict semantic identity (or at least exact identity in conventional English gloss), then "cow" matches "cow" and does not match "bull". Allowing sloppy semantics (as is inevitably necessary) creates vastly more degrees of freedom and more unfilled cells, which makes the null hypothesis most likely.
Similarly, comparison of letters has to be carried out with some kind of weighting, for example p is self-identical, is is highly similar to b, somewhat less similar to f, even less so with h, not much so with ŋ although there is a regular correspondence in Ugandan Nyole between Proto-Bantu *p and Nyole ŋ. The underlying premise regarding "chance" is that some linguistic relations are not the result of applying known genetic-historical linguistic processes to a common source word, instead the properties of the words (and any computed degree of similarity) are do to "something else" (which we call "chance"). Sound change can be context sensitive: therefore it is possible for p to correspond to both b and to ŋ, if the context is the same. When you add phonological context to the set of independent variables, you get a massive expansion of the number of cells to be filling, and again we have little reason to reject the null hypothesis.
We do at least have a weak basis for computing similarity of sounds, since any language sound can be assigned a "number" corresponding to the SPE phonetic features that describe the sound (though nobody uses that system for phonology anymore, still it is good enough for at least a first-pass comparison of "similarity"). With such a weighing of sound relations, you can meaningfully compare [p] and [b], and compute that [p] has little similarity to [ɮ], therefore is likely to not be due to ordinary historical change.