Switch? (The Monty Hall Problem)

Generalize the problem from three marbles to n marbles.

If there are n marbles, your odds of having selected the red one are 1/n. After
the other person selected a black one at random, your odds go up to 1/(n-1). 
There are n-2 marbles left in the bag, so your odds of selecting the red one
by switching are 1/(n-2) times the odds that you did not already select it
(n-2)/(n-1) or 1/(n-1), the same as the odds of already selecting it. Therefore
there is no advantage to switching.

If the person looked into the bag and selected a black one on purpose, then
your odds of having selected the red one are not improved, so the odds of
selecting the red one by switching are 1/(n-2) times (n-1)/n or (n-1)/n(n-2).
This is (n-1)/(n-2) times better than the odds without switching, so you
should switch.

This is a clarified version of the Monty Hall "paradox":

You are a participant on "Let's Make a Deal." Monty Hall shows you
three closed doors.  He tells you that two of the closed doors have a
goat behind them and that one of the doors has a new car behind it.
You pick one door, but before you open it, Monty opens one of the two
remaining doors and shows that it hides a goat.  He then offers you a
chance to switch doors with the remaining closed door.  Is it to your
advantage to do so?

The original Monty Hall problem (and solution) appears to be due to
Steve Selvin, and appears in American Statistician, Feb 1975, V. 29,
No. 1, p. 67 under the title ``A Problem in Probability.''  It should
be of no surprise to readers of this group that he received several
letters contesting the accuracy of his solution, so he responded two
issues later (American Statistician, Aug 1975, V. 29, No. 3, p. 134).
I extract a few words of interest, including a response from Monty
Hall himself:

   ...  The basis to my solution is that Monty Hall knows which box
   contains the prize and when he can open either of two boxes without
   exposing the prize, he chooses between them at random ...

   Benjamin King pointed out the critical assumptions about Monty
   Hall's behavior that are necessary to solve the problem, and
   emphasized that ``the prior distribution is not the only part of
   the probabilistic side of a decision problem that is subjective.''

   Monty Hall wrote and expressed that he was not ``a student of
   statistics problems'' but ``the big hole in your argument is that
   once the first box is seen to be empty, the contestant cannot
   exchange his box.''  He continues to say, ``Oh, and incidentally,
   after one [box] is seen to be empty, his chances are not 50/50 but
   remain what they were in the first place, one out of three.  It
   just seems to the contestant that one box having been eliminated,
   he stands a better chance.  Not so.''  I could not have said it
   better myself.

The basic idea is that the Monty Hall problem is confusing for two
reasons:  first,  there are hidden assumptions about Monty's motivation
that cloud the issue in some peoples' minds; and second, novice probability
students do not see that the opening of the door gave them any new
information.

Monty can have one of three basic motives:
1.  He randomly opens doors.
2.  He always opens the door he knows contains nothing.
3.  He only opens a door when the contestant has picked the grand prize.

These result in very different strategies:
1.  No improvement when switching.
2.  Double your odds by switching.
3.  Don't switch!

Most people, myself included, think that (2) is the intended
interpretation of Monty's motive.

A good way to see that Monty is giving you information by opening doors is to 
increase the number of doors from three to 100.  If there are 100 doors,
and Monty shows that 98 of them are empty, isn't it pretty clear that
the chance the prize is behind the remaining door is 99/100?

Reference (too numerous to mention, but this one should do):
    Leonard Gillman
    "The Car and the Goats"
    The American Mathematical Monthly, 99:1 (Jan 1992), pp. 3-7.