From: keith@celia.UUCP (Keith Goldfarb)
Newsgroups: sci.math,rec.puzzles
Subject: Re: Marilyn Vos Savant's error?
Date: 5 Jan 93 07:01:04 GMT
Reply-To: keith@rhythm.com (Keith Goldfarb)
Organization: Rhythm & Hues, Inc., Hollywood

Why do these simple probability problems always cause so much trouble?
This should all be in the FAQ's.  Perhaps we should even avoid discussion
of any English language problem.

If these problems are translated into any sort of rigid language, then
we would all be in agreement.  Why should we go on arguing with each
other when we are each solving different problems?

Having said that, here are my 2 cents about all of this:

    "A man walks up to you and says that he has two children, at
    least one of which is a boy.  What are the chances that the man
    has two boys?"

Now, the common interpretation of this is:

    From the space of all two-children families where at least
    one of the children is a boy, what is the probability that
    a randomly chosen family will be one with two boys?

What I'm getting at is that in the problem we aren't supposed to
take it at face value, but rather convert it to something we
can work with.  (After all, in the original text, the man's children
are already born, so either he has two boys or he does not, so the
"chances" are either 0 or 1, we don't know which!  But that's silly,
since it's the probability interpretation that is meant.)

In this case, things are pretty much in agreement.  But if we go further:

    "A man walks up to you and says that he has two children, the
    oldest one of which is a boy.  What are the chances that the man
    has two boys?"

Now we get into trouble.  Many people interpret this as:

    From the space of all two-children families where the oldest
    one of the children is a boy, what is the probability that
    a randomly chosen family will be one with two boys?

Which gives the often quoted result of 50%.  But some people don't 
think of it this way.  They don't see the man who has walked up to
them as making any "choice" from any "space".  Rather, he has
been blessed with two children, and he is particularly fond of 
telling us something about them.  Since we don't know why he
has chosen to tell us this information, we can't be certain of
the situation.  But a common and reasonable assumption is that
he is the type who likes puzzles and so he is telling us just
to ask us the probability.  This, like the Monty Hall problem,
can lead us to produce several different interpretations of his
motives and therefore of the event.

An interpretation of his motives that seems to be very common is:

    "A man walks up to you and says that he has two children, at
    least one of which is a boy.  He has the characteristic that
    if he had exactly one boy and one girl, he'd mention which of
    them (older or younger) was the boy.  But if he had two boys,
    he'd flip a coin to decide which one to mention, older or
    younger.  On this occasion, he says that the older child is a
    boy.  What are the chances that the man has two boys?"

This, interpreted, becomes:

    From the space of all two-children families where at least
    one of the children is a boy, a family is chosen at random.
    If the older child is a girl and the younger is a boy, then
    the trial is voided.  If both of the children are boys, then
    a coin is flipped and if it comes up heads then the trial
    is voided.  Of the trials which are not voided, what percentage
    of them occur with a family of two boys?

This gives the 1/3 result.  By modifying the man's strategy, you
can produce other probabilities.

Another way to look at this is that the reader has assumed that
the sample space will be "all two-children families where at least
one of the children is a boy" and then after a selection is made,
more information about that family is presented.  Some people
think of that additional information as reducing the sample
space, and some think of it simply as providing information
about a particular trial.  Without looking inside the speaker,
there is no way to determine which is the "correct" interpretation.

K.
-- 
Keith Goldfarb              Rhythm & Hues               Claudia is Everywhere.
keith@rhythm.com            celia!keith@usc.edu         ...mlogic!celia!keith  
I got nothing.  Too bad.                                Some pink toes
But I'm happy 'cause that's all I have.                 Some black toes