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I think you're missing that this columnist may in fact be a sensationalist hack. As far as I can tell, nowhere in Gott's paper does he imply that we will end in less than a thousand years. From a quick scan of the paper, Gott's most incendiary statement is that there is a 95% confidence interval that our remaining species lifetime is in the range of 12 to 7.8e6 years.

If anyone wants to check for themselves, hopefully it's not against HN rules to link the paper: https://sci-hub.tw/10.1038/363315a0

I think it's that you're unlikely to be alive at a time when only 100 humans had been born so you probably wouldn't be doing the calculation. Or, put another way, if you did do it then, you'd be in the minority of people who got the wrong answer. It's taking a sample of 1 (ourself) from the entire history of all humans. If that one person is randomly chosen, they're most likely to be alive in the fat middle of the population history, not at the thin tails (assuming there is a thin tail at the end?). I'm not sure how picking ourself is random though, maybe because we didn't decide when to be born?

You’re not missing anything, its just bad reasoning.