How likely am I to be descended from a particular person born in the year 1300?

Question

In other words, based on the following, what is p?

In order to make this a math problem rather than anthropology or social science, and to simplify the problem, assume that mates are selected with equal probability across the population, except that siblings and first cousins never mate, and mates are always selected from the same generation.

• $n_1$ -- initial population
• $g$ -- the number generations.
• $c$ -- the average number of children per couple. (If necessary for the answer, assume that every couple has exactly the same number of children.)
• $z$ -- the percentage of people who have no children, and who are not considered part of a couple.
• $n_2$ -- population at the final generation. (Either $n_2$ or $z$ should be given, and (I think) the other can be calculated.)
• $p$ -- probability of someone in the final generation being a descendant of a particular person in the initial generation.

These variables can be changed, omitted, or added to, of course. I am assuming for simplicity that $c$ and $z$ do not change over time. I realize this will get a very rough estimate, but it's a starting point.

Part 2 (suggestion for further research):

How can you consider that mates are not selected with globally uniform probability? In reality, mates are more likely to be of the same geographical area, socio-economic background, race, and religious background. Without researching the actual probabilities for this, how would variables for these factors come into play? How important would this be?

Answer 1

Because this question is receiving answers that vary from astronomically small to almost 100%, I would like to offer a simulation to serve as a reference and inspiration for improved solutions.

I call these "flame plots." Each one documents the dispersion of genetic material within a population as it reproduces in discrete generations. The plots are arrays of thin vertical segments depicting people. Each row represents a generation, with the starting one at the top. The descendants of each generation are in the row immediately beneath it.

At the beginning, just one person in a population of size $n$ is marked and plots as red. (It's hard to see, but they are always plotted at the right of the top row.) Their direct descendants are likewise drawn in red; they will show up in completely random positions. Other descendants are plotted as white. Because the population sizes can vary from one generation to the next, a gray border at the right is used to fill empty space.

Here is an array of 20 independent simulation results.

The red genetic material eventually died out in nine of these simulations, leaving survivors in the remaining 11 (55%). (In one scenario, the bottom left, it looks like the entire population eventually died out.) Wherever there were survivors, though, almost all the population contained the red genetic material. This provides evidence that the chance of a randomly selected individual from the last generation containing the red gene is about 50%.

The simulation works by randomly determining a survivorship and a mean birth rate at the beginning of each generation. Survivorship is drawn from a Beta(6,2) distribution: it averages 75%. This number reflects both mortality before adulthood and those people not having any children. Birth rate is drawn from a Gamma(2.8, 1) distribution, so it averages 2.8. The result is a brutal story of insufficient reproductive capacity to compensate for generally high mortality. It represents an extremely pessimistic, worst-case model--but (as I have suggested in comments) the ability of the population to grow is not essential. All that matters in each generation is the proportion of red within the population.

To model reproduction, the current population is thinned down to the survivors by taking a simple random sample of the desired size. These survivors are randomly paired (any odd survivor left over after pairing doesn't get to reproduce). Each pair produces a number of children drawn from a Poisson distribution whose mean is the generation's birth rate. If either of the parents contains the red marker, all the children inherit it: this models the idea of direct descent through either parent.

This example starts with a population of 512 and runs the simulation for 11 generations (12 rows including the start). Variations of this simulation starting with as few as $n=8$ and as many as $2^{14} = 16,384$ people, using different amounts of survivorship and birth rates, all exhibit similar characteristics: by the end of $\log_2(n)$ generations (nine in this case), there's about a 1/3 chance that all the red has died out, but if it hasn't, then the majority of the population is red. Within two or three more generations, almost all the population is red and will remain red (or else the population will die out altogether).

A survivorship of 75% or less in a generation isn't fanciful, by the way. In late 1347 rats infested with bubonic plague first made their way from Asia to Europe; during the next three years, somewhere between 10% and 50% of the European population died as a result. The plague recurred almost once a generation for hundreds of years afterwards (but usually not with the same extreme mortality).

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Code

The simulation was created with Mathematica 8:

randomPairs[s_List] := Partition[s[[Ordering[RandomReal[{0, 1}, Length[s]]]]], 2];

next[s_List, survive_, nKids_] := Flatten[ConstantArray[Max[#], 
   RandomVariate[PoissonDistribution[nKids]]] & /@ 
   randomPairs[RandomSample[s, Ceiling[survive Length[s]]]]] 

Partition[Table[
   With[{n = 6}, ArrayPlot[NestList[next[#, RandomVariate[BetaDistribution[6, 2]], 
        RandomVariate[GammaDistribution[3.2, 1]]] &, 
        Join[ConstantArray[0, 2^n - 1], ConstantArray[1, 1]], n + 2], 
     AspectRatio -> 2^(n/3)/(2 n), 
     ColorRules -> {1 -> RGBColor[.6, .1, .1]},  
     Background -> RGBColor[.9, .9, .9]]
    ], {i, 1, 20}
   ], 4] // TableForm

Answer 2

What happens when you try counting ancestors?

You have 2 parents, 4 grandparents, 8 great grandparents, ... So if you go back $n$ generations then you have $2^n$ ancestors. Let's assume an average generation length of $25$ years. Then there have been about $28$ generations since 1300, which gives us about 268 million ancestors at that time.

This is the right ballpark, but there is something wrong with this calculation, because the population of Earth in 1300 did not mix uniformly, and we are ignoring intermarriage within your ancestral "tree", i.e. we are double counting some ancestors.

Still, I think, this can lead to a correct upper bound on the probability that randomly chosen person in 1300 is your ancestor by taking the ratio $2^{28}$ to the population in 1300

Answer 3

The further back you go, the more likely that you are related to a person that successfully passed along their genes that lived in that time. Of the 1/4 billion ancestors that you have that lived in 1300, many of them would show up hundreds (if not thousands, millions) of times in your family tree. Genetic drift and the number of times we are directly related to someone are likely more relevant to the differences in our genetic code than who our ancestors were.

Answer 4

The probability is=1-z, every descendant in this problem is related to ancestors above. Whatever the initial rate of reproduction is (1-z) is your probability of being descendant from someone in the initial population.Only uncertain probability is what are the chances of being alive in final population.

I agree with Erad's answer, although I now think it responds to a question that was not asked - namely what is the probability that you are alive given certain known reproductive and population constraints on your fore-bearers.

Answer 5

My updated short answer is: $$ p > {(1-z)} \times {{{1} \over {n_1(1-z)}} \over {2}} = {2 \over n_1} $$

Answer explained:
Given a particular person today, it is certain that they are a descendant of at least 2 people in 1300.

When picking a particular person in 1300, there is (1-z) chance that person never reproduced, and the other term is for the number of 'parent couples', and the probability for the person to be related to this couple (1 / number of couples).

The (1-z) ends up cancelling out, leaving us with $$ p > {2 \over n_1} $$

Now just for fun but not necessary for solving the probability question
Here is the population of any given generation k in the chain between then and today. $$ n_{k+1} = {{n_k(1-z)\times c} \over 2} = {n_1(1-z)^kc^k \over 2^k}$$

Lets plug in some numbers as an example. For assumptions, I use:
g = 28 (25-year generations between 1300 and 2011)
n = 360M (world population estimate in 1300 from wikipedia)
z = 0.2, c = 2.77=8 (not real data, but does end up with about 7B people in 2011)

Resulting in:
$$p > 2 / 360,000,000 = 5.56 \times 10^{-9}$$ or over one in 180M.

Thanks for reading, Erad

Answer 6

This is a very interesting question as it is asking us to mathematically solve a fractal. Such as the famous game of life.

The % of the population which each generation related to will grow over each iteration, starting at $p_1={2 \over n_1}$ and at the limit generation will approach $\lim_{k \to \infty } p_k = (1-z)$.

If we denote $p_k$ as the probability of someone in generation $k$ to be related to the initial population. And for simplicity lets relax the siblings & cousins rule (can be added later). Then: $$p_1 = {2 \over n_1}$$

As each person in the new generation has exactly 2 ancestors in the initial population. $$ p_2 = relatives \times {2 \over n_2} + non.relatives \times {4 \over n_2} $$ In this case relatives could be calculated as: $$ relatives = {\binom{c}{2} \times {n \over c} \over \binom{n}{2}} = {c-1 \over n-1}$$ Or in other words, the number of sibling combinations, times the number of siblings family, divided by the total mating combinations. $$p_3 = immediate.relatives \times {4 \over n_3} + cousins \times {6 \over n_3} + non.relatives \times {8 \over n_3}$$

With each generation, the probability to be related to someone at the initial population will undoubtedly grow, but at a decreasing pace. This is because the probability to draw "relatives" which are coming from the same or similar tree will grow.

Lets use ethnicity as an example. Lets say we know for a fact someone is 100% Caucasian. At generation 28 he is most likely related to a significant portion of the Caucasian population in 1300 (As shown by @whuber simulation). Lets say he is marrying someone who is 100% of a different ethnicity. Their offspring will be linked to approximately double the number of people they are linked to from 1300.

Another interesting thought is that given the human (homosapien) race started from ~600 people in Africa, then we are most likely a genetic permutation of all of them who successfully mated.