In his article, "Investigations into the Doomsday Argument", Nick
Bostrom introduces the Doomsday Argument with the following example:
<< 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). >>
The Use of Unnumbered Balls
Let us first consider the case where the balls are not numbered. We
remove a ball from the left urn, and we wonder whether it came from
the urn containing ten balls or from the urn containing one million
balls.
The ball was chosen at random from one of the two urns. Therefore,
there is a 50% probability that it came from either urn. It is
important to realize that this probability is based on the number of
urns, not the number of balls in each urn, which could be any number
greater than zero.
There is nothing here to suggest a statistical limitation on the
maximum size of a group of balls.
The Use of Numbered Balls
Since the statistical limitation proposed by the Doomsday Argument is
not apparent with unnumbered balls, it may be a consequence of
numbering the balls.
The balls in the ten-ball urn have been numbered according to the
series of integers used to count ten objects (1, 2, 3, 4, 5, 6, 7, 8,
9, 10). The fact that each of these integers has been written on one
of the balls suggests that the balls have been counted in the order
indicated by the numbers. But if the balls had been counted in any of
numerous other different orders, the sum would have always been the
same, so the actual order used is of no significance.
Furthermore, if the physical distribution of the balls in the urn had
been arranged according to the series of integers written on the
balls, their distribution would not be at all random. If we imagine a
column of balls in each urn, ranging from 1 to 10 and from 1 to
1,000,000, the first ball selected at random from the two urns would
be numbered either 10 or 1,000,000. But we know from the statement of
Bostrom's example that the balls are arranged at random within the
urns.
Naming the Balls Uniquely
If the order in which the balls were counted is not significant, and
the balls have not been arranged physically in the order in which they
were counted, the numbers on the balls could still be used to identify
each ball uniquely, i.e., to give each ball a unique name. This idea
is supported by the fact that Bostrom wonders whether the ball 7
selected at random is the ball 7 from one urn or the other.
Because of the naming scheme used in the example, we could be certain
that any ball with a number greater than 10 came from the million-ball
urn. But the naming scheme has the flaw that it provides ambiguous
names for balls 1 through 10, which are found in both urns. It is, I
believe, this ambiguity in the naming of the balls that produces the
statistical result mentioned by Bostrom. The very same effect could be
produced by filling both urns with unnumbered white balls, except for
a single unnumbered blue ball in each urn. The two blue balls would
produce the same statistical effect as the two ball 7's.
If all of the balls had been numbered unambiguously from 1 through
1,000,010, the statistical effect produced by Bostrom's ambiguous ball
7 would vanish.
Gene Ledbetter
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Received on Mon Nov 26 2007 - 20:54:31 PST