If you'll indulge me, a thread. This week, I'm interested in ancestry. (1/n)
In particular: there's an apparent paradox when you think about family trees, which is this: (2/n)
You (almost certainly) had two parents. They each had two parents, so you had four grandparents. (3/n)
Follow it back, follow it back: eight great-grandparents, and generally if you go back n generations, your family tree has 2^n ancestors in it.(4/n)
Let's assume, just for simplicity, that a generation is 25 years. If you go back 400 years (16 generations), you had around 65,000 ancestors. (5/n)
If you go back 800 years, that's more like 4 billion ancestors alive... at a time when there were only about 400 million people on the planet. (6/n)
CONTRADICTION! Therefore aliens. (7/n)
Just kidding. You didn't have 4 billion ancestors alive. You had 4 billion spaces on that layer of your family tree. Those spaces were not necessarily filled by different people. (8/n)
At some point in the past, different branches of your family tree intertwined, so to speak. In other words, your parents are - most likely distantly - related. (9/n)
The question is: how distantly? How can we model that? (10/n)
Guys, it's time for some graph theory. (11/n)
I'm going to set up an imaginary graph, divided into generations. Each person is linked to two people from the generation above, their parents. (12/n) 
There are many, many false assumptions in this model, but let's roll with it. There are also many parameters that can be varied, but let's not. (13/n)
Instead, let's assume that the world population has grown by 9% each generation for as far back as we want to go, and that one's parents are randomly selected from the generation above. (14/n)
(This 9% number is wrong - it varies considerably - but it’s useful. It’s in the right sort of ballpark, at least.) (15/n)
In graph terms, how closely are two people related? We could quantify that by looking for the first common ancestor - and on average, how far back do we have to go to find them? (16/n)
Alternatively, we could save ourselves some bother by thinking about the hypothetical child of these two people. (17/n)
On average, how many generations (n) in their family tree must we go back so that the child is likely to have fewer than 2^n ancestors in that generation? (18/n)
My friends, we're looking at a variation of the Birthday Problem! (19/n)
There's some generation where we go from the kid’s number of ancestors being a power of 2, to it being... slightly less than a power of 2. (20/n)
With our assumptions, that's the same as picking 2^n people out of the whole population n generations ago, and them not all being distinct people. (21/n)
For a small number of ancestors and a large population, they're almost certainly all different. But as the number of ancestors grows, and the population shrinks, the chance of a collision increases. (22/n)
Now, we can look up the probability of k people, selected with replacement from a pool of N people, all being different: According to this: en.wikipedia.org/wiki/Birthday_…, it’s approximately...
If N_i is the number of people in generation i, counting backwards from the child, then the probability of their ancestors all being different in generation m is... in the picture. (24/n)
Ugh. Nobody wants to work that out. But we can work out the *logarithm* of p_m easily! It's in *this* picture. (25/n)
And when this value is less than ln(1/2), we have the value of n where we find our median common ancestor! (26/n)
What are the Ns? I'm going to say that N is the number of youngish adults alive — suppose 1/3 of the world is children, 1/3 of the world is more grown-up, and we have N ~ 2.5 billion (27/n)
Since each generation is 9% bigger than the last, I'll say that each previous generation is 8% smaller. (28/n)
That means we can simplify our estimate (with k=0.92) to this: (29/n)
A little Python script: repl.it/repls/Haunting… - one that I've not checked carefully - says that the first common ancestor would be about 15 generations ago, around the time of the English Civil War. (30/n)
I would expect the true answer to be somewhat closer to the present day - the assumption of 'all parents equally likely' is *way* off the mark - but this model gives a useful starting point. (31/31)
(cc @BarneyMT1)
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