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Re: A game strategy.



On Sun, 27 Jan 2002, Ludwik Kowalski wrote:

Play the game 50 times using paper cups as doors, a scrap of
red paper as a car and two scraps of white paper as goats. Being
the host (Monty) do not "follow any predictable patter", except
for one rule. That implied rule is that "the door hiding a car can
not be open before the second selection is made by the guest."
You will discover a significant difference between the outcomes
of two strategies: (a) switching the original selection and (b)
not switching the original selection. The first strategy, for a
large number of games, will yield twice as many winnings
as the second one.

Not true. Maybe you didn't read my first message carefully
enough. Once again, then. Suppose that Monty offers you the
chance to switch *only* in those cases where you have chosen the
door with a car behind it. What is your best strategy now? It
should take about a nanosecond to see (as I mentioned before) that
switching is *always* a bad idea in this case. If you do switch
you *never* win while if you don't switch you win 1/3 of the time.

Here's yet another scenario: Before you make your initial choice,
Monty decides whether or not he will subsequently open a different
door with a goat behind it and offer you a chance to switch to the
remaining unopened door. Half the time he decides he will and the
other half of the time he decides he won't. Your best strategy is
to switch whenever he offers you the chance to do so. By playing
this best strategy, you win half the time. ("Proof": Consider
playing six million games. Three million times you choose a door
and are not given a chance to switch. You win one million of
those times. Another three million times Monty opens a door with
a goat behind it and offers you the chance to switch to the
remaining unopened door. You always do so and, thus, win two
million times. Final results: 3 million wins out of 6 million
plays.)

You can also play the applet game but that would not be
convincing because we do not know if the underlying code
is a true simulation designed to validate (or invalidate) a
theory or a fake simulation designed to illustrate what is
"expected according to that theory." I did write codes for
true simulations of (a) and (b) strategies; they confirm the
theoretically expected 1/3 and 2/3 limits. It takes a second
to play more than 10,000 games on my six years old
computer. I will be happy to e-mail (or post) these easy
to follow True Basic codes, if somebody is interested.

You're preaching to the converted here. When I first heard about
the Monty Hall problem years ago, I too found it difficult to
believe the reported solution. Like you, I began to write some
code to simulate the game. But I never finished my program
because the simple process of writing it made it abundantly
obvious what the answer would be ... and why.

John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm