I love statistics, so I'm facinated by these two applications of statistics to the future of human civilization. First up, the Doomsday Argument, which uses statistics to demonstrate that humanity must be close to extinction.

The Doomsday argument was conceived by the astrophysicist Brandon Carter some fifteen years ago, and it has since been developed in a Nature article by Richard Gott [1993], and in several papers by philosopher John Leslie and especially in his recent monograph The End of The World (Leslie [1996]). The core idea is this. Imagine that two big urns are put in front of you, and you know that one of them contains ten balls and the other a million, but you are ignorant as to which is which. You know the balls in each urn are numbered 1, 2, 3, 4 ... etc. Now you take a ball at random from the left urn, and it is number 7. Clearly, this is a strong indication that that urn contains only ten balls. If originally the odds were fifty-fifty, a swift application of Bayes' theorem gives you the posterior probability that the left urn is the one with only ten balls. (Pposterior (L=10) = 0.999990). But now consider the case where instead of the urns you have two possible human races, and instead of balls you have individuals, ranked according to birth order. As a matter of fact, you happen to find that your rank is about sixty billion. Now, say Carter and Leslie, we should reason in the same way as we did with the urns. That you should have a rank of sixty billion or so is much more likely if only 100 billion persons will ever have lived than if there will be many trillion persons. Therefore, by Bayes' theorem, you should update your beliefs about mankind's prospects and realise that an impending doomsday is much more probable than you have hitherto thought.

Followed a decade later by Nick Bostrom's Simulation Argument.

ABSTRACT. This paper argues that at least one of the following propositions is true: (1) the human species is very likely to go extinct before reaching a “posthuman” stage; (2) any posthuman civilization is extremely unlikely to run a significant number of simulations of their evolutionary history (or variations thereof); (3) we are almost certainly living in a computer simulation. It follows that the belief that there is a significant chance that we will one day become posthumans who run ancestor-simulations is false, unless we are currently living in a simulation. A number of other consequences of this result are also discussed.

As Tyler Cowen points out, there are problems with both arguments since we have no reason to believe we are "random samples" of humans from which legitimate generalizations can be drawn.