This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
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?
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 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. Interviews with Monty Hall
indicate that he did indeed try to lure 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 occasionally Monty offered another door even when
the contestant had picked a goat. At any rate, analyzing the problem
with this strategy is difficult, since it requires knowing something
about Monty's probability of bluffing.
A good way to see that Monty is giving you information by opening doors
that he knows are valueless is to 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?
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).
However, the principles that underlie the problem date back at least to
the fifties, and probably are timeless. See the references below for
Reference (too numerous to mention, but these contain bibliographies):
Leonard Gillman, "The Car and the Goats", AMM 99:1 (Jan 1992), p. 3
Ed Barbeau, "The Problem of the Car and Goats", CMJ 24:2 (Mar 1993), p. 149
The second reference contains a list of equivalent or related problems.