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