From: zellner@stsci.edu
Subject: Re: Marilyn Vos Savant's error?
Organization: Space Telescope Science Institute
Distribution: na
Date: Sun, 3 Jan 1993 02:59:58 GMT

Steve White (srw@horus.ps.uci.edu) writes:

  > You forgot to label your table. You wanted it to be

  >    Older   Younger
  >    -----   -------
  >    Boy   -   Boy
  >    Boy   -   Girl
  >    Girl  -   Boy
  >    Girl  -   Girl

Correct.

  > Then your logic seems to hold. But if I label the table:

  >    Kid             Kid
  > Over there       Elsewhere
  >   -----          -------
  >   Boy         -   Boy
  >   Boy         -   Girl
  >   Girl        -   Boy
  >   Girl        -   Girl

Yes, you could of course do that, but it would be a different problem, and 
free of the apparent paradox:  Given that at least one of the kids is a boy, 
the probability of two boys depends on your knowledge of whether the one you 
see is the elder, the younger or simply "one of them." 

  >  Then case A gives probability 1/2 and case B gives 1/3.
  >  In other words, this argument fails. Neither wording is the
  >  same as "at least one of them is a boy".

Sorry, you lost me there.  If the man says "one of them is over there", and I 
look and see a boy, isn't that the same as "at least one of them is a boy"?

  >  The correct answer is 1/2 for A and B.

Try the following experiment.  Toss a penny and a nickel some large number of 
times, or just make up a table with equal frequencies of H and T for each, 
e.g., 

   Penny    Nickel
   -----   -------
     H       T
     H       H
     T       H
     T       T  

with equal frequency.  Let the penny represent the older child, the nickel 
the younger child, and heads or tails the sex of each.  Now go through and 
cross out all T-T combinations, that is, keep only those combinations in 
which at least one head (at least one boy) occurred.  Now ask the following 
questions: 

   1.  What is the probability of two heads, when the Penny is a head?
   2.  What is the probability of two heads, when the Nickel is a head?

Obviously the probability is 1/2 in either case.  But if you ask

   3.  What is the probability of two heads, when AT LEAST ONE COIN (of either
       species) is a head?

Clearly that's 1/3.  For a man to say "one of my two children is a boy" versus 
"the elder of my two children is a boy" samples two different statistical 
ensembles, with different results.

Now then:  Suppose a woman gets pregnant out of wedlock, and decides to give 
up the child for adoption straight from the delivery room, without even knowing
its sex.  A year later, she does it again.  But in later years the rascal who
kept getting her pregnant had a change of heart and married her and settled down
to be a good husband and father, and they began to wonder about those two 
childen. 

Information about adopted children is tightly guarded in some jurisdictions, so 
they hired a private detective to see what he could find out.  Suppose after
some investigation he reported: 

   "I've found the older child, and it's a boy."

Or instead,

  "I've found one of the children, and it's a boy, but the records I've been 
   able to peek at so far don't give its age, so I don't know whether it's
   the older or the younger."

In each of the two cases, what's the probability that both children are boys?  
It seems intuitive that the age of the child that he found wouldn't matter, but
mathematically it does.  That's the paradox. 

Cheers, Ben