Re: [Phys-L] two boy problems

[John Denker <jsd@av8n.com>]

On 10/28/2014 06:05 AM, Carl Mungan wrote:

> Providing additional positive information that is relevant
> will increase the fraction

> So apparently (and surprisingly to me) the hypothesis
> is false.

I take it the agenda item here is to make the correct result
less surprising.


1) The grid is the smart way to think about all such problems.

This grid is the origin of the term "marginal" probability;
if the grid is P(x,y) then P(y) gets written in the right
margin and P(x) gets written in the bottom margin.

Contrast this with much less clear methods such as we see
here:  http://en.wikipedia.org/wiki/Boy_or_Girl_paradox


2) Thinking in terms of /information/ is almost always a
good idea.  However, information is defined in terms of
probabilities.  If you get the probabilities wrong you'll 
get the information wrong, and vice versa.  In the present
example, it's not obvious how much /information/ is contained
in the various wordings.  It should raise a warning when
there are quibbles about what words are "relevant" or not.

So in this case, for me anyway, it is simpler and more
reliable to deal directly with the probabilities rather
than the information.

The key symmetry-breaker becomes more clear if we slightly
reword the statement to read:

   One child _(or both)_ is a boy born on a Tuesday.

The "or both" represents a collision, where a certain row
meets a certain column in the grid.  This happens in
boy/boy territory, never in girl/girl territory.  This
is key, because otherwise each row by itself is girl/boy 
symmetric.

==========================================

As emphasized by Pólya and others, after you have solved 
a problem such as this, you should think about it.  This
includes looking for generalizations.

In this case, it is amusing to investigate what happens if
we relax the requirement that the joint probabilities are
uniformly distributed.  The results are more sensitive to
this assumption than you might have guessed.  This leads
to Simpson's paradox ... which most people find vastly more 
surprising and confusing than the basic boy/girl question.

Simpson's paradox has serious implications for the design
of experiments, and the interpretation of experimental
data.

The wikipedia discussion is mostly OK and offers an
interesting baseball example:
  http://en.wikipedia.org/wiki/Simpson%27s_paradox

However, the statement "If weighting is used this phenomenon 
disappears" strikes me as completely wrong-headed.  The
phenomenon is real.  You don't want it to disappear.  By
using /improper/ weighting you can /conceal/ the phenomenon
from view, but it never really goes away.

Judea Pearl thinks clearly and writes clearly about such
things.  His careful but somewhat dry analysis is here:
  http://bayes.cs.ucla.edu/R264.pdf