I'm a "math guy" myself, although not a "trained mathematician", so let me throw this way of looking at it at you.
A chromosome pair is kind of like a railroad track. On one side, you have dad's DNA, on the other is mom's. So it's always 50/50, with mom & dad. The trick is, that the "rails" can switch - where mom was on one side, suddenly she's on the other, and vice versa for dad at the same spot.
At the same time that switch occurs, mom's DNA might switch from being grandpa's to being grandma's (or vise versa). These switches only occur, on average, a few times on a given chromosome, and they occur at fairly random places. So that means that on average ABOUT half of the DNA you got from mom is from one of her parents, vs the other - but NOT EXACTLY HALF.
So even if you could tell exactly which country every single base pair of your DNA came from, and if your 4 grandparents came from countries A, B, C, and D (and were "pure', ethnically), your results STILL wouldn't come out as exactly 1/4 each of A, B, C, and D.
As you go back further, say to your grandparents' grandparents, it only gets worse, of course. Instead of having exactly 1/16 from each, maybe one contributes as little as 4%, while another contributes 8%.
When you go back far enough, some of the ancestors might contribute nothing at all. So if you're 1/128 Swedish, due to that ancestor who lost the genetic game of musical chairs, your results wouldn't reflect your Swedish roots at all. If his wife was Native American, but you got double the average share of HER DNA, you'd come up as 1/64 Native American, even though you were really 1/128.
So even if they manage to pinpoint what country certain DNA segments come from (and it appears they can do surprisingly well with that sometimes) the percentages of each ethnicity being calculated can never, ever really be very accurate, no matter how much data is collected, or how clever they get with it.
Ethnicity percentages should ALWAYS be considered "for entertainment only", and never be taken seriously for real genealogy. It will get the continent(s) your ancestors came from right, but don't count on much more than that.
That being said, the test does have REAL genealogical information, regarding your biological relations, and that's the DNA matches that are provided. The quality of such matches is measured in centimorgans, and you generally don't have to worry about whether a given match might be a "false positive" unless it's below a certain level (that's pretty low). The main uncertainly there is in determining exactly HOW you might be related (2nd cousin, 3rd cousin, etc) from the centimorgans, and the higher the centimorgans, the more accuracy you get.