Ancestors and population paradox: exponential growth

If I have 2 parents, and each of them have 2 parents, each generation back has double the number of members. Thus there are 2^n members n generations ago. 10 generations ago there were about one thousand (1024) people, 20 back were 1 million, and 30 back were 1 billion.


If each generation is about 25 years, then 30 generations ago is around 750 years ago, or around 1250 AD. The entire world population is estimated to be around 300 million people around 1300 AD.


This implies that the number of people in my family tree 30 generations ago (1 billion) is larger than the number of people who existed (0.3 billion).


Does that means that I am equally likely to be descended from everyone who lived 700 years ago?


We’re all cousins!


If my wife and I have 2 kids, and they each have 2 kids, and they each have 2 kids….


My dad, a human geneticist from Honolulu, is visiting me in Seattle. He (and my mom) broke the two kids per family rule: he had five boys.


Comments (16)
  1. Todd Ostermeier says:

    Too many factors to easily compute this way. For example, what if one woman had a child each with two different men? Those children would share an ancestor, and would make it three people in the previous generation rather than four. Think about polygamy, harems, etc, where one man may have children on many different women. If the group is large enough, that would equal little more than one person in the previous generation (the mother) for each child, since the father would factor out of the equation (see the large portion of Asian people with DNA that can be traced back to Ghengis Khan — he was prolific enough to essentially wipe himself out of the equation).

  2. Rob says:

    30 generations back there were an unknown number of family members because the ones you calculated were still not born yet. The number you calculated must be the number of members at this moment of time if no one died during the 30 generations. I like to meet your 750 year old grandfather! If someone is married 6 times or they have 10 childeren doesn’t influence calculations. The real factors are diseases like the "Black disease" killing 1/2 of earth population at ones.

  3. WildFire says:

    Some of us are from outer space.


  4. RyanNerd says:

    This is an example of how statistics can be stacked to arrive at an incorrect conclusion. For instance, using statisical calculations I can prove that ice cream causes rape. As the number of rapes in a city goes up so does the number of ice cream sales the charts are almost identical for frequency and time. Why is this??? Because most rapes occur in the summer and ice cream sales increase in the summer as well 🙂

    Frankly, I’m just glad that the only silly comment so far is about space aliens and not something serious like the flood during Noah’s time that would have knocked the population down to 8 souls to start all over again 🙂

  5. Sean says:

    You are assuming that every male and female in your 30 generation family tree appear only once.

    Assume that we all descend from Adam and Eve. They would appear at the end of every line you trace back. The 2^30 ancestors = 1,073,741,824 would be 25% Adam and 25% Eve and the rest would be their descendants.

    The real situation is that we all have ancestors that appear in both the male and female sides of our family trees, (hopefully) many generations back.

    The number of separate individuals will be far less than 1,073,741,824.

    In a thoroughbred horse pedigree of 1,022 ancestors (not 1,024) there will be about 400 different individuals. Some may appear in a pedigree 10+ times. Inbreeding = close duplications; linebreeding = 5th, 6th generation duplications and further back.

  6. drebin says:

    Your also missing another key element – ALL of those people weren’t alive all at one time.

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  9. Tanveer Badar says:

    Your reasoning is seriously flawed. You seem to be under the ‘delusion’ that all those people were distinct, ruling out the trivial cases like cousin marriages.

  10. blake says:


    I am struggling with this notion of not enough ancestors for the decendants. I undersatnd cosuins marrying cousins but in the end are we all traced back to a few common ancestors? So in the end we all are related? What did that person meanthat some of us come form space. Does DNA testing show how many common ancestors there are in the world?

  11. says:

    I think the original post was totally valid mathematically, and raises two very significant questions – how much intermarriage between "cousins" has been taking place during the evolution of the human race, and to what extent have population control, and mass deaths due to wars, plague etc. managed to contain the planet’s population at its present level?

    Obviously, in early tribal societies, the total available living population spanning, say, three generations, would have to be drawn on to maintain the tribe, and inbreeding would have been inevitable.  Equally obviously, therefore, the mathematics cannot reflect the reality.

    Does anyone know of a site that contains some real insight into the "paradox"?  I would be interested to know.

  12. Further to my previous post, I did find an interesting site at "

  13. Alex says:

    This exponential theory is fundamentally flawed – it assumes that all couples will have 1 child.

    If this was the case humans would get extinct. You would need 10 trillion people 1000 years ago to have a population of 6 billion today which after a generation would be 3 billion (as 2 will only create 1 desendant)and after 1000 years there would be no one left. (numbers not precise – only used to illustrate the point).

    Instead the population is always increasing because people have more than one child so form a population of 300 million 1000 years ago we are now 6 billion. (again rough numbers)

    To make it really clear, think of it this way, if a couple has 10 children then the 10 children will have come from 2 parents – according to the flawed logic of this article they would have come from 20…

    Its really simple, no aliens or inbreeding…

  14. Andy the right says:

    This doesn't actually have anything to do with how many children anyone has, Alex. this is about Everyone having two parents. It doesnt matter how many siblings.The math is correct. no the numbers do not include the whole tree from then to now, we are talking about exponential tiers of generations of ancestors. this is about following lineage, for example your paternal grandfathers tree, plus your maternal grandfathers tree. Also it doesnt take very many generations back for the number of the tier to exceed the population of the planet at the time. we are not talking about isolated tribes here, we're talking middle ages. If there were some prolific Abrahams, then we are higher percentages of them than of our others

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