# Check My Math on Number of Ancestors by a Naive Calculation

So Denis Lamoureux may have found an inconsistency, and maybe even an error in my book. I write on p. 47:

For a population of one million, for example, the IAP would be at about thirty-five generations. Using a generation time of twenty-five years, Using a generation time of twenty-five years, that is less than 900 years ago.

So, I’m using a generation time of 25 years here.

For example, there are about 160 generations between 10,000 and 5,000 years ago. Naively (and falsely) assuming there is no overlap in our family trees, we can compute the number of ancestors alive 10,000 years ago from the population at 5,000 years ago, 18 million people; we arrive at about 2.6 x 10^55 ancestors. This is more ancestors than the number of stars in the visible universe. However, there were just 2 million people alive 10,000 years ago. How is this possible? The ratio between these numbers is 1.3 x 10^49. This is how many times ancestors at 10,000 years are being double counted in this naive calculation, and it is an astronomical number of times.

So here is the problem. Denis rightly notes that 5,000 / 25 is 200, so where did I get 160 generations? As I recall, I was using a generation time of 30 for the calculations in that paragraph (which give’s us 167 generations in 5,000 years), but I do not say this. So we have an inconsistency.

Seems I have two options.

1. Change the first calculation to use a generation time of 30.

2. Change the second calculation to use a generation time of 25.

Seems like option 2 is the best, but that requires calculating all those numbers again. I don’t want to make a mistake here again…so I’m going to check my math closely this time…

Okay, using:

g = generation time = 30
p5 = population at 5 kya = 18M
p10 = population at 10 kya = 2M

We arrive at:

5000 years / g = 167 generations
estimate_p10 = p5 * 2^{5000 / g} = 2.67x10^57 people
estimate_p10 / p10 = 1.34x10^51 ratio

So that means even if we use generation time of 30, there are two errors:

2.6 x 10^55 should be 2.7x10^57
1.3 x 10^49 should be 1.34x10^51

So what if we use a generation time of g = 25?

5000 years / g = 200 generations
estimate_p10 = p5 * 2^{200} = 2.89x10^67 people
estimate_p10 / p10 = 1.45x10^61 ratio

It seems that these are the numbers I should use to keep things somewhat simple in the text, by using a single generation time instead of multiple. So, I expect to be issuing an erratum that will be changing the second paragraph to:

For example, there are about 200 generations between 10,000 and 5,000 years ago. Naively (and falsely) assuming there is no overlap in our family trees, we can compute the number of ancestors alive 10,000 years ago from the population at 5,000 years ago, 18 million people; we arrive at about 2.9x10^67 ancestors. This is more ancestors than the number of stars in the visible universe. However, there were just 2 million people alive 10,000 years ago. How is this possible? The ratio between the estimated and known population is 1.5x10^61. This is how many times ancestors at 10,000 years are being double counted in this naive calculation, and it is an astronomical number of times.

Yes, I think that’s the best approach (and kudos for correcting yourself).

I did a very similar calculation in this blog post, with similar results: Mitochondrial Eve and Y-chromosomal Adam – Bravus

"We all have a vast number of ancestors. There’s a conundrum here. I have two parents, 4 grandparents, 8 great-grandparents, 16 great-great-grandparents and so on. It’s a sequence going up in powers of 2: I’m 2^0, my parents 2^1, grandparents 2^2, great-grandparents 2^3 and so on.

If we assume a human generation is about 20 years, that’s 5 generations a century, 50 a millennium. Perhaps 50 x 150 = 7,500 in 150,000 years. But the thing is, 2^7500 is 5.3 x 10^2257. There are only about 7.5 x 10^9 people on Earth right now, and that’s the most there’ve ever been: certainly not that other outlandish number."

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