[Thought experiment about 20 numbered people]
>> Are you aware that this is the same reasoning that gives rise to
>> the Doomsday argument? Do you accept that argument? If not, why?
Not exactly. The problem is properly specified. What "gives rise" to the Doomsday Argument is that it is not properly specified, so one must make assumptions to fill in the gaps--and different people make different assumptions.
For example, "probability" exists only as a relationship between a set of events and a superset ("universe") of events. Both must be specified for probability to have any meaning. In the problem above, the universe of events is clear--you might be one of 20 given people. The subsets of events in whose probability we are interested is well-defined too, so we can answer it. In the DA, the universe of events is "all possible physical universes"--a concept so ill-defined that one could write pages just on what that means.
The second assumption, of course, is self-sampling. That too has more problems in the DA than here, because in the DA we must make some assumption about just what we are a sample of, whereas in the problem above that's a given.
Finally, the "100!-th digit of pi" bit is irrelevant--it's no different from a poker player using probability to determine the proper bet on the next turned card, even though its identity is already strictly determined by an earlier shuffle, although with the slightly added twist that the frequency of digits is not strictly random (but then again, neither is a shuffled deck of cards).
-- Lee Daniel Crocker <email@example.com> <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