[battyboy12]

I am an anthropology student from Michigan. I am currently researching the inter-relatedness of most, if not all, humans alive today. Anyway, I have a personal interest in mathematics, statistics and probability theory and have incorporated it into my studies. I am not looking for answers from an anthropological or historic perspective, I'm just looking for answers from a mathematical/statistical one. Any information if my theory is in any way "valid" is what I am seeking.

Before I elaborate my theory, here is some background information:

According to the Law of Truly Large Numbers... "that with a sample size large enough, any outrageous thing is likely to happen.In a sample of 1000 independent trials, the probability that the event does not happen in any of them is

, or 36.8%. The probability that the event happens at least once in 1000 trials is then 1 − 0.368 = 0.632 or 63.2%. The probability that it happens at least once in 10,000 trials is

.
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:

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 too 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, due to this probabilistic law, almost everyone alive today had at least one ancestor from like say, China or Japan.
Would this theory likely be true given enough time?

Once again, please not I'm looking for anything from a anthropological or historical perspective. I already have that part of the research down. I just would like an educated opinion from a statistical standpoint. Thank you for your time and efforts.

[joypulv]

"Lets say 500 years ago, due to this probabilistic law, almost everyone alive today had at least one ancestor from like say, China or Japan.
Would this theory likely be true given enough time?

There are excellent mathematicians here. I'm not one, just interested in this.

I wonder if you could be more exact about your theory? Your first statement states a premise that's really a question, and it's based on math, but it's based on math that doesn't factor in real population patterns.
It would seem to me that to calculate when 'enough time' will be, you'd have to go back to a time when we really did all have at least one ancestor from one place, namely back 70,000 years?

[ebaines]

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

I don't agree. You have a 1/500 chance that one of your ancestors is the same as one of my ancestors. When you multiplied this by 1,000,000 you get a certainty that at least out of 1 million randomly selected people today at least one pair have a common ancestor. But that does not mean that it's a certainty that any randomly-selected pair today have a common ancestor - that is already shown to be abut 1/500.

Lets say 500 years ago, due to this probabilistic law, almost everyone alive today had at least one ancestor from like say, China or Japan.
Would this theory likely be true given enough time?

If mates are truly random then yes. But they're not. The chance of a 15th century woman from, say, England marrying a man from, say, Japan was nill. It would have been rare for a couple to marry who lived more than 20 miles apart, much less 7000 miles. So choosing your spouse was not random, but was highly correlated to where you lived. Until the 20th century technical and economic barriers to travel severely limited options for finding a spouse from outside your village. And of course even today religious and ethnic traditions further limit ones choices for marriage. So the idea that one's ancestors are randomly distributed is not at all realistic.