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Topic: interesting probability problem
Replies: 9   Last Post: Oct 17, 2007 3:20 AM

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Robert Israel

Posts: 3,262
Registered: 2/1/07
Re: interesting probability problem
Posted: Oct 16, 2007 12:25 PM
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Andersen <> writes:

> Bertsekas has the following exercise in his probability book:
> "
> Consider a statement whose truth is unknown. If we see many examples
> that are compatible with it, we are tempted to view the statement as
> more probable. Such reasoning is often referred to as inductive
> inference (in a philosophical, rather than mathematical sense). Consider
> now the statement that “all cows are white.” An equivalent statement is
> that “everything that is not white is not a cow.” We then observe
> several black crows. Our observations are clearly compatible with the
> statement, but do they make the hypothesis “all cows are white” more
> likely?
> To analyze such a situation, we consider a probabilistic model. Let us
> assume that there are two possible states of the world, which we model
> as complementary events:
> A : all cows are white,
> C(A) : 50% of all cows are white. [C(A) is the complement event of A].
> Let p be the prior probability P(A) that all cows are white. We make an
> observation of a cow or a crow, with probability q and 1−q,
> respectively, independently of whether event A occurs or not. Assume
> that 0 < p < 1, 0 < q < 1, and that all crows are black.
> (a) Given the event B = {a black crow was observed}, what is P(A|B)?
> (b) Given the event C = {a white cow was observed}, what is P(A|C)?
> "
> ---
> Solutions to a) is p, and b) is 2p/(1-p). From this he draws the
> conclusion that a) does not affect the hypothesis A, while b)
> strengthens it. Is this reasoning correct? I mean event B should have
> the same effect as event C, as B supports A, since A is equivalent to
> "everything that is not white is not a cow". What is wrong in my reasoning?

Look up "Hempel's paradox". There are many alternative scenarios.
In many of them, observing anything other than a cow does support the
hypothesis that all cows are white, because it favours states of the world
in which the number of cows in the world is smaller, and if there are
fewer cows there is less chance that an anomalous colour will appear.
For example, consider the following two possible states of the world.

(W_1) There exist 10 cows and 10^7 crows.
(W_2) There exist 10^7 cows and 10 crows.

In each case, suppose the probability of a given cow not being white is
1/10^6, independent of everything else. So in W_1 the probability
that there exists a non-white cow is only about .00001, while in
W_2 it is over .99995. If we select a random object from the world
and it turns out to be a crow, this makes W_1 much more likely than W_2,
so it increases the probability that all cows are white.
Robert Israel
Department of Mathematics
University of British Columbia Vancouver, BC, Canada

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