Re: [Phys-l] Monty Hall problem

[John Denker <jsd@av8n.com>]

Here's a parable in the form of a dialog:

  Salvatio:  I have here a small multi-faceted object.  I toss the 
  object onto the table.  I observe that in this instance comes to 
  rest with a certain facet on top, and there are six dots on that 
  facet.  Now suppose I toss it again.  How many dots will be on
  the top facet?

  Simplicio:  Obviously six.  You just told me that when you toss
  it, there are six spots on top.  There is nothing else I need
  to know.  Let's make a game of it.  Every time it comes up "six"
  you pay me a dollar.  If it ever comes up non-"six" I will pay
  you a hundred dollars.

  Sagredo: I expect Simplicio is going to lose a lot of money.  The 
  fundamental problem is that given one element drawn from a 
  distribution, he thinks he can infer the underlying probability.

  [they play the game]

  Simplicio:  I forgot to tell you:  I assume you are required to
  toss the object in such a way that it comes up "six" every time.
  That's the rules.

  Salvatio:  You don't get to change the rules after the game has
  started. 

On 01/06/2011 07:08 AM, chuck britton wrote:
> Perhaps Marilyn's explanation was wanting but her response to the
> problem as stated in her column cannot be slighted. Her version(s)
> always included the host opening a goat door and offering a switch.
> Whether he had angelic or devilish intent is entirely irrelevant to
> the problem(s) as given in Parade magazine.

That is false and fallacious many times over.

The fundamental problem is that Marilyn has given us *one element*
drawn from some distribution.  It is observed that in this one
instance, the host opened a goat-door.  Note the contrast:
 -- Concerning this one instance, we agree that the host's 
  intent has got nothing to do with it.  The fact is that in
  this one instance, he did open a goat-door.
 -- Concerning the underlying probability distribution:  the
  rules that govern the host's behavior matter a great deal.
  They make the difference between losing 100% of the time
  and winning 100% of the time (or anything in between) if
  you follow Marilyn's advice.

I don't know how to say it more plainly:  In the real world you
cannot infer the distribution of probabilities from a single
instance.  Marilyn published a single instance, and drew from
it probabilistic conclusions.  This involved making assumptions.
These assumptions do not reflect real life;  they are simply
assumptions pulled out of some orifice.  Neither the original
question she published nor the original answer she published
even said what assumptions she was making.

  On the other hand, in *subsequent* publications she did
  say what assumptions she was making.  This however does not
  change the fact that her original unrestricted answer was 
  quite wrong, and her subsequent restricted answer is not
  the answer to the original question.

  The added restrictions are an example of changing the rules
  after the game has started.

Let's be clear:  
 a) Marilyn's advice is to always switch.
 b) That is not the correct answer to the original, unrestricted
  question.