How many ancestors?

How many ancestors do you have? If you go back some number of generations, how many ancestors are there in that generation?

Assuming you're the product of normal human reproduction (clones can hit the "back" button right now), you have two parents. They had two parents, so you have four grandparents -- four ancestors if you go two generations back. Going back three generations, you have eight ancestors; four generations, sixteen ancestors; and so on. In mathematical terms, if you go back n generations, you have 2ⁿ ancestors.

Not!

This line of argument becomes preposterous if you carry it back far enough. Go back ten generations and you have 2¹⁰ = 1024 ancestors. Call it 1000 just to make things easier. If you go back another ten generations, each of those 1000 has 1000 ancestors, so twenty generations back from you there are 1,000,000 ancestors. Thirty generations back there's another factor of 1000, so you have 1,000,000,000 ancestors -- one (American) billion.

Which can't be right. Thirty generations back is roughly 1000 years -- and there weren't a billion people on Earth then. (See "How Many People Have Ever Lived on Earth?") Granted, not all the people in the thirtieth generation back from you lived at the same time, but that doesn't help much.

And it gets worse and worse. Ten more generations back and you have a trillion ancestors. This is about ten times more than the estimated number of people who have ever lived. Ten more generations, a quadrillion ancestors. And so on.

Yet everyone has two parents. So the only way you can avoid this ridiculous conclusion is to realize that some of your ancestors must have had ancestors in common.

In a simple case, if your father's grandparents on his father's side are the same people as your mother's grandparents on her father's side, then you have only six ancestors three generations back, not eight: Your father's maternal grandparents, your mother's maternal grandparents, and two more people who are the paternal grandparents of both your parents. To put it another way, there are two different paths back from you to two of your great grandparents.

But if so, then your parents are first cousins. And generally, if you have ancestors who had ancestors in common -- as you now know you must -- then that means those ancestors were related. Somewhere in your ancestry, you have couples who were some sort of relatives of one another.

Well, we knew that, of course. We all must have common descent from some individual ancestor, whether one of the earliest modern Homo sapiens of 50,000 years ago, or an earlier species of Homo a million or more years ago, or some other primate even earlier. And that means all couples in your ancestry (and everyone else's) were relatives of one another.

But the above argument proves you must have multiple paths back to the same ancestors in much more recent times. Your parents probably have common ancestors within the past 1000 years -- and so, probably, do you and your spouse, or you and your best friend. I myself know of several cases of multiple paths back from me to ancestors in the 18th century -- some a little more scary than others! Of course if one person's ancestors all came from (say) Easter Island and another's all came from Scandanavia, their common ancestor is likely to be further back. Maybe even then, not as far back as you'd think. Douglas LT Rohde has written a paper (PDF format) reporting on a series of computer simulations which suggest all people in the world today have a most recent common ancestor (MRCA) who lived about 2000 to 5000 years ago. (Go a few thousand years further back and everyone on earth at that time who has any descendants alive today is an ancestor of everyone alive today!)

Anyway, if you have any northern European (especially English or Danish) ancestry... then you and I probably have a common ancestor who lived after, say, 700 AD. Jack Lee argues that he knows who that ancestor is: Charlemagne. (Or, less flippantly, that everyone living at the time of Charlemagne who has any descendants alive today is an ancestor of everyone of European descent alive today.) His argument is more informal than Rohde's and I would take it with a grain of salt, but on the other hand it's hard to argue that the probability he estimates for not being a descendant of Charlemagne (10^-15,000 (!)) can be wrong by 14,999 orders of magnitude.

Here's another thought, based on reading Richard Dawkins's The Ancestor's Tale and having more to do with biology than genealogy. An obvious truth once you think about it (at least, obvious to anyone who understands evolution well enough to know it is proven fact), but perhaps surprising when first pointed out anyway. Start with the fact that not only are all humans related to one another, but we're all related to all other organisms -- our MRCA with clams may have lived hundreds of millions of years ago, but there was one.

So now think about this: Somewhere in the past several million years there lived an individual who was the MRCA of all living humans and chimpanzees. He (or she, I'll use "he") was my ancestor, your ancestor, and Washoe's ancestor. In principle, a child (which I'll call "she") of this MRCA could have been ancestor to some but not all humans and some but not all chimps; but the above arguments suggest she was ancestor to all humans or none -- and likewise ancestor to all chimps or none. In any case, if we shift our attention to the most recent individual who was an ancestor of one or more living humans and one or more living chimps -- who may have been that same MRCA or may be a later descendant -- then by definition, he must have had two or more children of whom at least one is ancestor to one or more living humans (very probably all living humans), but no chimps, and at least one is ancestor to one or more (probably all) living chimps, but no humans. Think about that. Once upon a time there were, without doubt, two siblings, probably growing up together, all the living descendants of one of whom are humans, and all the living descendants of the other of whom are chimpanzees!

This of course doesn't mean one sibling was a human and one was a chimp; they both were members of the same ancestral species, very much like one another, and had their life histories varied a little, the human ancestor could have ended up being the chimp ancestor and vice versa. An intelligent observer would have had no way of knowing they were at the divergence point of future species. But they were.

Similarly, go back far enough and there were two siblings, all the living descendants of one of whom are mammals, and all the living descendants of the other of whom are birds or reptiles. Or siblings whose living descendants are vertibrates for one, invertibrates for the other. Or siblings whose living descendants are on the one hand, animals, and on the other hand, plants...

And just think, maybe a few hundred million years from now the descendants of you and your siblings will be just as different!

How many descendants?

Working forward is similar to working backward. The difference is that we all have two biological parents, but we have varying numbers of children. The world's population generally increases with time, but fairly slowly, which means the average number of children (that survive to adulthood) per person (who survives to adulthood) is larger, but not much larger, than 2. Of course it varies from time to time and place to place, but overall it must be a little over 2.

In the following, let's assume the average is exactly 2, for convenience. The number of descendants n generations removed is (on average) 2ⁿ, which is equal to the (exact) number of ancestors n generations removed -- if there is no intermarriage between relatives, and of course there is.

But that's an average. Some people have no children -- or have children none of whom have children, or grandchildren none of whom have children, or whatever -- and so have no descendants after some number of generations. Others have many.

Here's a question prompted by thinking about mitochondrial DNA (mtDNA), which is inherited only along the female line: yours comes from your mother, which comes from her mother, which comes from her mother, and so on. Yours will be inherited by all your children if you're female, or by no one if you're male, and will (if you're female) be passed to the children of your daughters, but not the children of your sons, and to the children of the daughters of your daughters, but not the children of the daughters of your sons nor the children of the sons of your daughters, and so on. What is the probability of your mtDNA being passed on to the (n+1)th generation after you? That is, what is the probability that there will be a female line extending n more generations after you? On the face of it, it might seem pretty unlikely after only a dozen or so generations: any female line will end if a woman has no daughters, and that's not very improbable. But female lines of twenty or more generations are actually not very unlikely.

Of course female lines do get more and more unlikely as the number of generations increases. After all, whatever the probability of all female lines ending by a certain generation is, the probability of ending by the next generation has to be higher. But how rapidly does the likelihood of a female line shrink?

That depends on two things: the distribution of the number of children you and your descendants have, and the probability that a given child is female. (Again, by "children" I mean children who survive to adulthood.) That probability isn't exactly 1/2 but is close, and we can take it as exactly 1/2 without much error, at least for most times and places.

(I think intermarriage probability enters into it too, but for now assume no intermarriage.)

The distribution of the number of children is another matter. It's hard to know how to model that realistically. So let's not. Let's instead consider two models: (A) Exactly two children per person, that is, probability of 2 children P(2) is 1.0, and (B) Probability of 0 (1, 2, 3, 4) children is 0.1 (0.2, 0.4, 0.2, 0.1). In case B the average number of children is 2.

Then the probability of having no daughters is sum (P(n) * (1/2)ⁿ). In case A this is just 1 * (1/2)² = 1/4. In case B, .1 * 1 + .2 * 1/2 + .4 * 1/4 + .2 * 1/8 + .1 * 1/16 = 0.33125, somewhat larger, even though the average number of children in both cases is the same as in case A. Of course the probability of having no sons is the same as the probability of having no daughters. The probability of having at least one daughter is 1 - 1/4 = 3/4 in case A, 1 - 0.33125 = 0.66875 in case B.

What's P_d(m), the probability of having (exactly) m daughters? In case A, P_d(0) = 1/4, P_d(1) = 1/2, and P_d(2) = 1/4. In case B, P_d(0) = 0.33125, P_d(1) = 0.4, P_d(2) = 0.2125, P_d(3) = 0.05, and P(4) = 0.00625. Trust me on these, or compute them yourself.

Now, if we know the probability P_nf(g) of having no female line descendants after g generations, then what is P_nf(g+1) of having no female line descendants after g+1 generations? It is the sum over m of (probability of having m daughters * probability of all m daughters having no female line descendants after n generations), that is, sum (P_d(m) * (P_nf(g))^m). In case A,

P_nf(g+1) = 1/4 + 1/2 * P_nf(g) + 1/4 * P_nf(g)²

Here is P_nf(g) for g from 1 to 20 in the two cases:

g	A	B
1	0.25	0.33
2	0.39	0.49
3	0.48	0.58
4	0.55	0.65
5	0.60	0.69
6	0.64	0.73
7	0.67	0.76
8	0.70	0.78
9	0.72	0.80
10	0.74	0.81
11	0.76	0.83
12	0.77	0.84
13	0.79	0.85
14	0.80	0.86
15	0.81	0.87
16	0.82	0.87
17	0.83	0.88
18	0.83	0.88
19	0.84	0.89
20	0.85	0.89

So even after twenty generations there's about a 10% to 15% chance the mtDNA is being passed along.