Re: MATH: Bayesian story problem

From: Eliezer S. Yudkowsky (sentience@pobox.com)
Date: Tue Mar 25 2003 - 11:56:44 MST

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    Lee Daniel Crocker wrote:
    >
    > I'm with Spike; I got 5/8 and didn't do anything Bayesian.

    Well, the *first* part of that is right... but take, for example, the
    step in your reasoning where you say:

    > Since 25% of the eggs are blue with pearls, and 40% are blue,
    > then 5/8 of the blue eggs have pearls

    Bayes' Theorem is sometimes written:
    p(A|X) = p(A&X) / p(X)

    (which is of course what

    p(A|X) = p(X|A)*p(A) / [p(X|A)*p(A) + p(X|~A)*p(~A)]

    simplifies to)

    In the way Bayesian problems are usually presented, you take the prior
    probability p(pearl) and the two conditional probabilities p(blue|pearl)
    and p(blue|~pearl), three numbers with three degrees of freedom, to arrive
    at p(pearl|blue). But actually if you have *any* three probabilities from
    the set of 16 probabilities p(pearl), p(~blue), p(~blue&pearl),
    p(~blue|~pearl), p(pearl|~blue), et cetera, and the three probabilities
    have three degrees of freedom among them (you can't derive the third
    quantity from the first two), then it suffices to derive the whole set.
    The story problem is just a dramatic illustration of this.

    A verbal solution:

    Suppose you have a large barrel containing a number of plastic eggs. Some
    eggs contain pearls, the rest contain nothing. Some eggs are painted
    blue, the rest are painted red. Suppose that 40% of the eggs are painted
    blue, 5/13 of the eggs containing pearls are painted blue, and 20% of the
    eggs are both empty and painted red. What is the probability that an egg
    painted blue contains a pearl?

    If 40% of the eggs are painted blue, 60% are painted red.
    If 60% of the eggs are red, and 20% of the eggs are red and empty, then
    40% of the eggs are red and contain pearls.
    If 5/13 of the eggs containing pearls are painted blue, 8/13 of the eggs
    containing pearls are painted red.
    If 40% of the eggs are red and contain pearls, and 8/13 of the eggs
    containing pearls are painted red, then 65% of the eggs contain pearls.
    If 65% of the eggs contain pearls, and 5/13 of the eggs containing pearls
    are painted blue, then 25% of the eggs are blue and contain pearls.
    If 40% of the eggs are painted blue, and 25% of the eggs are blue and
    contain pearls, then 5/8 of the blue eggs contain pearls.

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


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