I was talking with a friend yesterday and he asked me to give an answer to this question: How many people have ever lived? I had no clue how to answer this question or to even where to begin to answer this question. After a feeble attempt at estimation, I asked him how he would do it and he gave me a pretty clever answer.

First he made some assumptions:

1. Humans have been living for 200,000 years

2. Humans grow similar to bacteria, they double every generation

3. The average generation over the past 100,000 years is 50 years

4. There are 10 billion people alive today (very liberal assumption)

After establishing these assumptions, he begun his calculations. If humans have been living for 200,000 years and a generation lives for 50 years, there have been 4,000 generations. We have also assumed that the population doubles. Therefore if there are 10 billion today, there were 5 billion people last generation.

We can setup a sum of 4000 terms to calculate the number of people that ever lived…

(10B + .5*10B +.25*10B + … .5^(3,999)*10B)

For approximation sake, we can treat this as an infinite sum, and simplify the calculation to

a /(1-r) = 10B/(1-.5) = 10B *2 = 20 Billion people have ever lived

After browing the web for answers other people have gotten, the number seems to be somehwere in the hundreds of billions so our estimate is potentially way off. I think one assumption that needs to be modified is the growth rate which has been variable over different time priods. But I think that these back of the envelope calculations are important, and, with the right assumptions, could give us a quick idea of answers to such questions. I wonder if anyone else can come up with a better way to estimate this number?

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‘100 billion’ is relatively close to ’20 billion’ in terms of ‘order of magnitude’.

One thing occurs to me though: You don’t end up using the 200 000 years assumption, or the 50 years assumption. What you’ve done when you make the sum infinite is to get rid of the ‘4000’ from the calculation entirely. In fact, in your final equation, you really just have a and r, which is

a = 10B = people today

and

r = 1/2 = ratio of people in each generation to the next generation

So you only use your second and fourth assumptions in the final calculation! A little simpler.

It seems the generational growth rate is all important (is it really 2:1?), and as you point out, this has probably varied greatly. If the growth rate is lower, then you get a higher number. So probably assuming each generation doubles (over the history of the species) is the part that needs adjustment, if the estimate is off.

There are some other interesting observations to make, too.

One is that in this model, it doesn’t matter much when we stopped being erectus and became sapiens, since that affects only the tiny ‘tail’ of the infinite sum.

Another is that there’s no real reason to round off 7 billion to 10 billion since the final calculation is just 2a: the total number that ever lived just doubles today’s population. So your model would predict 14 billion people.

From a look around the net, this seems to be the standard reference, and it takes an approach based on the growth rate per year:

http://www.prb.org/Articles/2002/HowManyPeopleHaveEverLivedonEarth.aspx

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Prof. Kateon October 21, 2008at 10:26 pm