## 2022/01/22

### Doomsday argument

And now for something completely different: a fun little probability puzzle.

What's the probability that the human race will end some time in the next 100 years? Surprisingly this question has a logical answer. And not because we have some magic crystal ball.  In fact, our puzzle  specifically assumes we have no information at all about the future.

Here's how it goes.  Let's step out of time for a second, and consider the total number of humans who will ever exist. Let's say that number is N.  If you are of the Abrahamic faiths, you can call the first one Adam. But we're just having fun so we'll just number them from first to last: 1,2,3, ..., N.  Now let n be your number. So 1 <  n < N, you are somewhere between the first  and the last person ever.  Since we  have no information about the future, we have no clue if you are near the end or near the beginning or somewhere in the middle.  You just happened to land at some random position in the long line of  humans. So we have to assume that any position is equally likely, or technically that n is uniformly distributed between 1 and N. The chance that you are in a particular interval is equal to how big that interval is relative to the whole sequence. There's a 50% chance that you are in the first half and 50% chance that you are in the second half,  there's a 95% chance that you are in the first 95% and a 5% chance that you are in the last 5% of people, etc.  So P(n < f*N) = f and P(n>f*N)  = 1-f, for any fraction f between 0 and 1.

We don't know N, but we can estimate n, because we can approximately calculate the cumulative population to date. This is more accurate  than you might think because the parts really long time ago where we have poor estimates are also the times where there were very few people.  The left tail is long but thin. Estimates now are around  n = 117 billion.

From the above, the distribution of N is P(N<n/f) = 1-f. That means there's a 5% chance that N < 123B i.e. that there are only 6 billion babies to go before the last one. If we translate that into time, using the current rate of 140M births per year,  it means there's a 5% chance that we have less than 43 years left! And a 50-50 chance that we'll be around for another 800 years. At the other end, a 5% chance that we have more than 16,000 years left, and so on.

I heard about this puzzle known as the "doomsday argument" about a year ago. Of course you can debate about whether this is a realistic model, but it's a cute way to provoke thought about all the minor risks we collectively worry about and the big ones we don't consider rationally.

Reminds me of a few scenarios discussed in this blog a long time ago:  ineffective posturing on climate changethe asteroid lottery , political pandering in a pandemic... Ouch ouch ouch! Sadly humanity doesn't seem to have gotten wiser in the decade (!) since those posts... 43 years seems like an awfully short time. At least math is eternal!