Re: A simple betting problem was RE: my objection to the Doomsday argument

From: Eliezer S. Yudkowsky (sentience@pobox.com)
Date: Wed Apr 30 2003 - 13:25:49 MDT

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    Rafal Smigrodzki wrote:
    >>
    >>Here's another trick question that only the true Bayesian will
    >>resolve. You meet a mathematician. "How many children do you have?"
    >>you ask. "Two," he replies, "and at least one of them is a boy."
    >>What is the probability that they are both boys?
    >>
    >>I would *not* answer 1/3.
    >
    > 1/2 ?

    That's one defensible answer for a Bayesian.

    The classic form of the brainteaser runs as follows. You meet a
    mathematician, who says he has two children. You ask "Is at least one of
    your children a boy?" and the mathematician answers yes. In this case, of
    course, the chance that the remaining child is a girl is 2/3.

    If you meet a mathematician who just happens to remark "At least one of my
    children is a boy", it is obvious that he is trying to pose the above
    brainteaser. We can suppose a prior 1/4 chance that he has two boys, a
    prior 1/4 chance that he has two girls, and a 1/2 chance that he has a
    boy-girl or girl-boy pair. If he has two boys, he must pose the
    brainteaser by saying "At least one of my children is a boy." If he has
    two girls, he must pose the brainteaser by saying "At least one of my
    children is a girl." We suppose that if he has a mixed pair, he is
    equally likely to pose the question one way or the other. If the
    mathematician says "At least one of my children is a boy", the
    probabilities to consider are p(.25)*p(1) and p(.50)*p(.50). So it is
    equally probable that the mathematician's other child is a boy or a girl.

    But if I encountered this situation in real life, I'd guess around an 80%
    chance that the mathematician had a mixed pair. Why? Because the
    mathematician knows that the mixed pair is the "correct" answer, and would
    be less likely to choose this problem as a good illustration if that were
    not actually the case... 

    -- 
Eliezer S. Yudkowsky                          http://singinst.org/
Research Fellow, Singularity Institute for Artificial Intelligence

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