From: Eliezer S. Yudkowsky (sentience@pobox.com)
Date: Fri May 11 2001 - 19:30:51 MDT
Lee,
I don't think your word problems provide the necessary priors. To
estimate the probabilities, I would need to know the algorithms used by
each of the people making statements, and that information is not
provided.
For example, suppose that you're on a game show called "ExtroQuiz" and
there are three doors: A, B, and C. One door has a prize behind it. You
pick door C. The game show host opens up door A, and shows you that it's
empty. What is the probability that the prize is behind door B? Answer:
0%. Why? Because the game show host *knows* the standard answer to this
riddle, *knows* that most Extropians will switch to B, and he opens A *if
and only if* the prize is actually behind C.
The "mathematician's daughters" is a much worse instance of the problem.
If you know that the mathematician will say "At least one of my children
is a girl" for BG, GB, and GG, but say "At least one of my children is a
boy" only for BB, and the mathematician says "At least one of my children
is a girl", then the probability that the other child is a boy is 2/3. If
you know that the mathematician will say "At least one is a girl" for GG,
"At least one is a boy" for BB, and pick a statement at random for BG and
GB, then the probability that the other child is a boy is 1/2. I think
that most people unconsciously assume this kind of symmetry by default -
it seems, a priori, a more likely visualization of the underlying causes -
which is why most people return the allegedly "wrong" answer 1/2. To
assume otherwise requires that one assume the mathematician has an
inherent bias towards naming girls, which is an odd thing to assume about
someone. The above mixture of probabilities is exactly what results from
the mathematician picking one child at random and saying "At least one of
my children is an X", in which case the statement provides no information
about the second child. Most people unconsciously assume that's what the
mathematician did, in which case the given answer, 1/2, is exactly
correct.
I've *always* felt that riddle had a problem; now, twelve years later, I
finally get to articulate it.
(In Lee Corbin's question as stated, I expect that the mathematician
hypothesizes the father to be symmetrical, rather than hypothesizing that
the father exhibits a feminine bias, and answers 1/2.)
-- -- -- -- --
Eliezer S. Yudkowsky http://singinst.org/
Research Fellow, Singularity Institute for Artificial Intelligence
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