### Which two are more genetically different... two randomly chosen humans alive today? Or a human alive today and a direct (paternal/maternal) ancestor from say 10,000 years ago?

Disclaimer: I'm not an expert in this and speak with no authority on genetics or biology.

As annoying of an answer as this is, the best I can do is "it depends."

You have 2 parents, 4 grandparents, etc. doubling each generation. Each parent contributes exactly 50% of your DNA, but because of how sexual meiosis works, you're probably not getting exactly 25% representation from each grandparent. One of your regular cells has 50% mom's and 50% dad's DNA, but one of your sperm/egg cells has 50% of your total DNA; and since 50% of your total DNA comes from each parent, it's mathematically possible but statistically pretty much impossible to have a sperm/egg that's nearly pure mom or dad. A sex cell is fairly close to 25/25 most of the time, but it's technically possible that after a few generations you wouldn't have inherited any DNA directly from one of your ancestors. As you start talking about double digit generations, it's certain that some ancestors are contributing more than others.

Further, if you wanted to know how many 39 x great grandparents you have, you'd take 240 and find that you should have had about 1.1 trillion 39 x great grandparents, which is about 11 times the total number of humans that have ever lived. And that's only going back 40 generations, or about 1000 years. You've got a whole lot of repeats in there. That amplifies the problem I just mentioned where some individual branches of your family tree supply a lot more of your DNA - the same guy could be your 39 x great grandfather 1000 times! The further back it goes, the more billions and billions of slots you have to fill with the same people over and over again. For example, all modern humans of European descent are descended from a group of seven women who lived tens of thousands of years ago. So if you're a European or European American, your 1000th generation of female ancestors is a string of those seven women, with their names repeated octillions of decillions of gigagoogols of times.

As far as modern day humans, there's a 99.999% chance that you are 16th cousins or less with anyone in the world. For example, Barack Obama is Cheney's 8th cousin and Bush's 11th cousin. For populations that are more ethnically contained or isolated, like tribes that haven't contacted the outside world or a small island nation like Iceland, you're going to see a lot more "repeats" in their family tree going back even a few generations.

So what's the answer? Who do you share more DNA with - your 8th cousin down the street, who shares some unknown amount of DNA from a couple 9 generations ago, or your prehistoric grandma, who appears in your family tree somewhere between 1 and 2100 times? I don't know, and I don't think anybody else does either.

TL;DR: It's an interesting question without an easy answer. Each ancestor doesn't contribute equally to your DNA. Somebody of your race in your country is almost certainly your cousin going back a single digit number of generations, but because of meiosis and probability it's impossible to say how much DNA you would you share.