Exponential Growth on a Family Tree

Date: 12/26/2002 at 22:38:29
From: Sam
Subject: Exponential Growth on a Family Tree

I was wondering about the probability of my being related to a famous 
person of the past. So I was looking at how many people I am descended 
from depending on how many generations I go back.

I looked at the my family's family tree (which is very shallow, only 
going back to the 1700s), and I noticed that for each generation I 
went back the number of people doubled.

I recognized this as exponential growth. It should be represented by 
the equation y = 2^x

The difference in age between generations varies widely, gradually 
increasing as we get into modern life spans. I approximated that the 
difference between generations was about 20 years.

Therefore 2000 years ago was 2000/20 = 100 years ago
y = 2^100
Which gives me y = 1.2676506 E 30
or
1,267,650,600,000,000,000,000,000,000,000

But my answer is unreasonable. That is far more than the number of 
people who have ever lived! What did I do wrong?

Date: 12/27/2002 at 10:46:13
From: Doctor Shawn
Subject: Re: Exponential Growth on a Family Tree

Sam,

What a great question! However, you have to think about it this way: 
if you're going to assume that in each previous generation the
population doubled, then eventually the human population will keep
halving until there are only two people left, and then they will die
out. But this is clearly wrong, because there are a lot of people
alive now and more all the time.

The answer to your puzzle is called "pedigree collapse." Basically,
you have a bunch of slots on your family tree, but some of them are
filled by the same people. For example, in some tribes in the Amazon
it's common for people to marry their first cousins. Normally, a
person has eight great-grandparents. However, someone who marries his 
or her first cousin only has six individuals as great-grandparents, 
because two of them are duplicates. (This might be a little 
complicated, but if you do a chart it will become clear.)

This is something of an extreme example, but it does happen in real
life. Alfonso XIII, King of Spain, only had 10 great-great-
grandparents, for example. For other examples of how this
works in real life, check this out:

   Pedigree Collapse - John Becker
   http://www.generations.on.ca/genealogy/pedigree.htm 

Now most people in our society don't marry their close cousins, but
the principle still holds. Some geneticists estimate that no one on
earth is more than one fiftieth cousin from anyone else. Although that
level of relationship is very distant, everyone on earth is related to
everyone else in SOME degree. And that means that no one can have
children without having some duplication in ancestors. If you're
interested in this topic, you should read "The Mountain of Names" by
Alex Shoumatoff. Your local library ought to have it.

Good luck!  Write again if you'd like to talk about this some more, or
about anything else.

- Doctor Shawn, The Math Forum
  http://mathforum.org/dr.math/