LC> Would the fact that we find ourselves numbering less than
> 100 billion truly give us reason to suspect we are among the
> unfortunate slated to perish soon? That depends upon whether
> or not the question is being asked by someone who _fairly
> chose our civilization to pick on_ by offering this bet.
NB> In a sense you are betting against yourself when you decide
> how to plan your life in view of your estimates of how long our
> species will survive.
Precisely; which is why you are not an acceptable random sample. If we are betting a short-lived civilization against a longlived one, then we must offer the bet to a random sample of ALL the postulated possible civilizations, not just the one at hand. You can't have it both ways: either other civilizations are postulated, in which case we are not a good sample, or else we are a good sample, in which case we may not postulate others.
Bayes' requires an initial P(H), and P(E|~H). The former doesn't matter much; it can be vanishingly small and still generate a final probability as high as 1/3 in my examples. But the latter--the prior probability of finding the evidence conditioned on the negation of the hypothesis--i.e., how likely are we to find our birth order small if we are a sample from the postulated civilizations other than ours-- is critical. A large final result requires this prior to be large; and the larger this is, the poorer a sample we are.
-- Lee Daniel Crocker <firstname.lastname@example.org> <http://www.piclab.com/lcrocker.html> "All inventions or works of authorship original to me, herein and past, are placed irrevocably in the public domain, and may be used or modified for any purpose, without permission, attribution, or notification."--LDC