<pre>
Under reasonable assumptions about Monty Hall's motivation, your chance
of picking the car doubles when you switch.

The problem is confusing for two reasons:  first, there are hidden
assumptions about Monty's motivation that cloud the issue; and second,
novice probability students do not see how to handle the information
that the opening of the door gave them.

Monty can have one of three basic motives:
<pre>
 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 car.
</pre>

In each case, the information from Monty opening the door is:
<pre>
 1.  The probability of the car distributes evenly over both remaining doors.
 2.  The probability of the car in your door does not change.
 3.  The probability of the car in your door goes to 100%.
</pre>

These result in very different strategies:
<pre>
 1.  No improvement when switching.
 2.  Double your chances by switching.
 3.  Don't switch====
====
</pre>

Most people think that (2) is the intended interpretation of Monty's
motive.  However, even given this interpretation, many people at this
point still have trouble with this puzzle.  Here are two other
arguments that sometimes do the trick.

1. Increase the number of doors from three to 100.  If there are 100
doors, and Monty shows that 98 of them are valueless, isn't it
pretty clear that the chance the prize is behind the remaining door
is 99/100?

2. All agree that you have a 1/3 chance of picking the car
originally.  That means that the other two doors represent a 2/3
chance.  Now, after Monty opens the one of them that is a goat
(which he must always be able to do assuming he knows which one it
is), the remaining door must now represent the 2/3 chance.  So you
should pick it.

In the real game show the situation was much more complex.  Interviews
with Monty Hall indicate that he sometimes lured the contestant who had
picked the car with cash incentives to switch.  However, if Monty
always adopted this strategy, contestants would soon learn never to
switch, so one presumes that Monty offered another door even when the
contestant had picked a goat.  At any rate, analyzing the problem when
Monty was luring people away from the car is difficult, since it
requires knowing something about Monty's probability of bluffing.
See the paper by Fernandez and Piron for an analysis of this game.

The original Monty Hall problem (and solution) appears to be due to
Steve Selvin, and appears in American Statistician, "A Problem in
Probability," Volume 29, Number 1 (February 1975), page 67.  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 in American Statistician, Volume 29, Number 3 (August
1975), page 134.  However, the principles that underlie the problem
date back at least to the fifties, and probably are timeless.  See the
references below for details.

These references are selected from the voluminous literature on this
subject because they are current and contain bibliographies:

Leonard Gillman, "The Car and the Goats," American Mathematical
Monthly, Volume 99, Number 1 (January 1992), page 3

Ed Barbeau, "The Problem of the Car and Goats," College Mathematics
Journal, Volume 24, Number 2 (March 1993), page 149. Contains a list of
equivalent or related problems.

Luis Fernandez and Robert Piron, "Should She Switch? A Game-Theoretic
Analysis of the Monty Hall Problem," Mathematics Magazine, Volume 72,
Number 3 (June 1999), page 214
</pre>
