In the midst of World War 2, there was an effort by the allies to determine the number of German Panzer V tanks in production. It was observed that the serial numbers for the wheels of their tanks were ordered sequentially, based on when they came off the production line. With knowledge of the serial numbers of the tanks that had been captured or destroyed prior in the war, the Allies were able to generate estimates and confidence intervals for the total number of tanks in production each month. These were the statistical estimates for three different months, along with a comparison to conventional military intelligence estimates at the time¹:

Month	Statistical estimate	Intelligence estimate	German records
June 1940	169	1,000	122
June 1941	244	1,550	271
August 1942	327	1,550	342

   The statistical estimates proved much more accurate than the conventional military intelligence. The actual calculation is pretty straightforward. If we assume there are a total of m tanks around, and we’ve recovered k of them, then we might expect the k serial numbers to be evenly distributed between [1, m]. You can imagine the k samples being evenly spaced throughout the sample: 1 –– x₁ –– x₂ –– … –– x_k –– m. So intuitively our estimate for the maximum value m would be one more even-spacing [(x_k – k)/k] past the highest serial number we obtained: m ~= x_k + (x_k – k)/k = x_k(1 + 1/k) – 1 ²

   Now, using this on German tanks is cool and was actually useful in real life.

   How about we instead go for awesome but useless? 😈

   If we apply the same sort of statistical reasoning instead on our own life being an N=1 sample from the population of humans that will exist throughout time, along with some assumptions on human population over time, we can also get an estimate (and confidence bounds) of when humanity itself will go extinct!

   This is known as the Doomsday Argument, or Carter catastrophe. It was originally proposed by astrophysicist Brandon Carter in 1983³, and subsequently championed by the philosopher John A. Leslie.

   To use Leslie’s numbers⁴: if we estimate that 60 billion humans have been born so far, then we could argue that there is a 95% chance that we are not within the first 5% of humanity, and so less than 20*60 billion = 1.2 trillion humans will exist throughout time in total. If we then assume that the world population stabilizes at 10 billion and with a life expectancy of 80 years, then the remaining 1.14 trillion humans will be born within the next 9120 years. And so, we conclude that there is a >95% chance that humanity goes extinct within the next 10,000 years.

   But of course, there are caveats. N=1 is a small sample size, and are we really sampling from the population of humans? Could we include animals, cyborgs, or aliens in our ‘reference class’ of things we could have been born as? Shouldn’t we update our estimate with other information, e.g. reducing it given the relative rarity of observed extinction-level events throughout history?

   There are wackier (in my opinion, poor) possible objections too^5,6.

   But I have in the past wondered about the “odds” I was born as a human, and not as an ant, despite their difference in number. Perhaps the odds of ‘being’ a thing is correlated to the level of its consciousness? Certainly I think there was no chance “I” were to exist as a rock, and probably not as a carrot either.

   Of course, when I say the “odds” of “me” “being” a “thing”, so much may be lost in the fuzzy words, that perhaps I am saying nothing at all. 🙂

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1. An Empirical Approach to Economic Intelligence in World War II
2. Estimating the Size of a Population
3. The anthropic principle and its implications for biological evolution
4. A Bayesian Analysis of the Doomsday Argument
5. The “self-indication assumption” argues that your own probability of existing is a function of the number of humans born when you were out of how many could have been, and should not be assumed to be one. https://philsci-archive.pitt.edu/2144/
6. Some have tried instead to apply the doomsday logic using the lifespan that theorems exist before being refuted (if they ever are) to the Doomsday Argument itself, to result in a paradox. http://philsci-archive.pitt.edu/1205/1/gott1f.pdf