The Law of Truly Large Numbers & Ancestors

[bejee44]

I'm an anthropology student. I'm currently studying the inter-relatedness of most- if not all- human beings alive today. I also have an interest in statistics so I am using probability theory to come up with estimates. Here is something I go

This is just for my own theory- something I can up with. According to the Law of Truly Large Numbers... "that with a sample size large enough, any outrageous thing is likely to happen.

This law was made by Persi Diaconis and an associate

.
This means that this "unlikely event" has a probability of 63.2% of happening if 1000 chances are given, or over 99.9% for 10,000 chances. In other words, a highly unlikely event, given enough tries, is even more unlikely to not occur."

That was the Law of Truly Large Numbers. What I have below is from a computer scientist- Douglas Rohde- in regards to common ancestors between any random two people alive today:

If there were random intermixing, then we would each have ~1 million ancestors living in 1500 AD, out of a world population of ~500 million.
So the fractional overlap between two people would be about 1/500th.

But the probability that two people share at least one common ancestor would be essentially 100%. Basically, you are choosing a random number between 1 and 500 a million times and you're asking whether you ever choose number 500. In a million trials, we expect this to happen 2000 times. So that it happens at least once is guaranteed.

If we get rid of the random intermixing, the fractional overlap will drop to much less than 1/500th. But I suspect that the probability of at least one overlap will remain very high.
If the population in 1500 was 500 million, and it is 6 billion today (12x larger).
If the average generation length is 30 years, there are 17 generations in 500 years.
So the average number of surviving children per mother is exp((log 12)/17) = 1.157
Since a child has two parent, the average number of surviving children per person is 2 * 1.157 = 2.315
So this is the average growth rate per generation for the descendants of a person in 1500.
2.315^17 = 1.575 million. So an average person in 1500 has about 1.5 million offspring alive today. Sampling from the whole world, the probability that a random person from 1500 is an ancestor of a random person in 2000 would be 1.5 million / 6 billion = 0.025%.
If you were only considering people in a region like Europe, it would probably be something like 1.4 million / 700 million = 0.2%
Here is my theory:

 

Lets say 500 years ago or so, due to this probabilistic law, almost everyone alive today had at least one ancestor from like say, China or Japan*, or almost whatever remote region that had at least some contact with other places.

 

*please note Japan was not totally isolated at this time

Would this theory likely be true given enough time?