Part 2
If we go back 10 generations, 116*(210), we see that as many as 118,784 segments are in play. If we use an approximate average of 7,200 calculated centiMorgans per sex-averaged genome and say that a value even as low as 5cM can accurately be detected (I would raise that, but most-optimistic scenario here), then our genomes would house about 85,556 detectable segments. You see the conundrum. Segments don't neatly divide during meiosis like, say, sheets on a roll of toilet paper (World's Worst AnalogiesTM), but just the raw numbers demonstrate how ancestral genetic material drops out over generations. Some small chunks live on due to linkage disequilibrium, but much of the DNA sifts out over time so that our distant cousins are unlikely to have much in common with us, genetically. We end up patchwork quilts made from quite different bits of cloth.
Trying to validate at the 8th cousin level means, assuming no recent pedigree collapse, you have 512 paper-trail ancestors and your cousin also has 512 paper-trail ancestors...but you supposedly share only two among those 1,024 potential genetic donors.
I keep saying "recent pedigree collapse" because those effects--barring actual endogamy and oft-repeated incidents of it--filter out very quickly. For example, if you and I were both 3rd cousins and 4th cousins, the difference in the amount of expected DNA (theoretically) would be only about 0.196% (call it about 14cM) than it would be if we were 3C only. That drops in half with the next generation, our kids, assuming no further pedigree collapse. By our grandkids, the amount is too low for our standard tests to reliably measure, around 3cM.
We have some pretty good, peer-reviewed analyses of the likelihood that two cousins will share any detectable DNA (Henn at al., 2012; Caballero, et al., 2019). For 7th cousins, that's about 1% of the time; 7C1R, about 0.57%; for 8th cousins, about 0.29%. By those numbers, we'd need to test around 345 8th cousins to find two who had matching, measurable, DNA.
But wait. There's more math! How great is that?!
Those percentages denote any matching DNA. Not the same bit of DNA that's identifiable as having come from the same, identified ancestor. Let's round up the 8th cousin matching and state it as a probability: 0.003. Because any two children of the assumed MRCA would share only about 50% of the same DNA, we can basically divide that probability in half to arrive at an approximation of the probability that any two 8th cousins will share some of the same DNA that originated from the same identified ancestor: 0.0015.
This next bit will not be accurate...because, biology! I've tried off and on for about three years to interest any of several well-known bioinformaticians to take up the challenge, but no luck so far. It's complicated. There's more in play than just straight math, but that's all we have to work with for now. Remember our "independent event" where the outcome of one event doesn't have any effect on the outcome of another? To arrive at a probability involving something like a throw of the dice, we'd simply multiply the separate probabilities.
That would mean the probability of finding three 8th cousins who all shared the same measurable segment of DNA that came from the same ancestor would be somewhere around 0.0015x0.0015. That's a sort of staggering 2.25x10-6...or a 0.000225% chance. The reality could easily be a factor of magnitude higher, but that would still mean we'd be testing over 44,000 8th cousins to locate three who meet the criteria. Not exactly lotto-winning odds, but not great.
It's such a humongous outlier that, if we think we've found it, Occam's razor would tell us that the proposed solution to the DNA sharing that's being displayed is far, far too complicated to be the best explanation. Part of the scientific method for any good lab experiment is to actively, diligently work to disprove the hypothesis. That's the only way to try to eliminate confirmation bias and to gain confidence that the hypothesis will stand up to rigorous scrutiny.
Unfortunately, popular genetic genealogy has led us to believe that it works the other way around: establish the hypothesis (A and B both descend genetically from C) and then set out to "prove" it. And that's also an unfortunate disconnect: genealogy uses the word "proof" liberally, as in the Genealogical Proof Standard, and that was an established practice long before traditional genealogy ran headlong into the biological sciences and bioinformatics a couple of decades ago.
On the science side, there is really no such thing as "proof" (as I've used recently, that's why Einstein's theory of general relativity is still labeled a theory even though it's been tested time and time again since 1911). In the physical and life sciences you can't "prove" a hypothesis, you can only disprove it.
For genetic genealogy we should be working more like a scientist and less like a genealogist. Gather all the data possible that might be germane to the hypothesis; then try to gather more data; then work to disprove the hypothesis, i.e., find other explanations for the data; and be sure to attempt to reconcile all data that's still left hanging, at least where it's possible to do so.
What I see far more often with autosomal triangulations is that the process starts at the wrong end of the proposition and stops when any data at all seems to present itself as a possible correlation to the hypothesis. E.g, I think A and B both descend genetically from C. I go run a triangulation at GEDmatch. Lo and behold, even though GEDmatch has a database only about 6% the size of AncestryDNA's, I find a bunch of reported possible triangulations that include that segment and, oh look!, there's B right there among the 28 other people listed for A as mutually sharing that segment. Done! I've "proven" that A and B descend C with a triangulated segment of DNA.
But, nope. Nothing "proven." In fact, nothing even analyzed yet. Apparent correlation does not equal causation. Takes a lot of detailed work.
I'm here through the weekend with two shows Friday and Saturday night. Please be sure to tip your waitstaff. They work hard and your gratuity is a big part of their income. Thank you all! You've been great audience!
<cough cough>