CAR!GOAT  --  a probability simulation in Pygmy FORTH

                   by David Zethmayr

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The Three-Door CAR!GOAT probability poser is a lovely quasi-
paradox you can read more about in the file GOAT-CAR.DOC on
the Educational RT.  It can be stated like this:

     You are the contestant in a Game Show where the
     challenge is to choose the winning Door out of three
     closed doors, exactly one of which hides the prize: a
     new Car.  Each of the other two doors hides a Goat
     (a non-prize).

     After you announce your original choice, the Game Show
     Host opens one of the other two doors, revealing a
     goat.  (This, by the way, is the way the game always
     proceeds, and everyone knows the procedure.)

     After opening a Goat door of his choice, the Show
     Host offers you a second choice, namely to switch from
     your original choice to the other closed door.

     The question, then, is this: is it of any advantage to
     you to switch?

Test your answer with the program CAR!GOAT, written in
Pygmy FORTH for XT/AT/386 MS-DOS computers.  You are the
contestant, the program is Game Show Host and Tallykeeper
as you play multiple times.

The Tallykeeper tracks the times you switched and the times
you didn't, tallying the wins for each case.

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Here is my attempt at explaining/predicting the results.

  Explanation
  ===========
Consider your second choice--switch or no--from two cases
at that point:

Case 1:  picked one of the Goat doors originally.
    What is the probability you will now win by switching?
    ( it is certain, or 1)

Case 2:  picked the prize door originally.
    What is now the probability of winning by "keeping"--
    not switching? ( certain-- 1)

There are no other cases from the point of view facing the
second choice.

Now consider the original choice.

At the outset, with what probability will your original
choice position you for a certain win by switching?  It's
exactly the probability of choosing a Goat door originally.
(2/3)

This is your overall expectation of winning by switching.

At the outset, with what probability will your original
choice put you in position of a certain win by "keeping"--
not switching?  It's exactly the probability of choosing
the Car door originally.  (1/3)

This is your overall expectation of winning by keeping.

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A FORTH course may be starting soon in CALC, GEnie's
Computer Aided Learning Center.

CAR!GOAT was written in Pygmy FORTH version 1.3, a program-
ming environment well suited to learning FORTH, as well as
serving its primary purpose as a superbly crafted engineer-
ing tool.  You can get Pygmy by hopping over to the FORTH
RoundTable on GEnie and downloading the latest version.
Frank C. Sergeant is its author.  Please don't fail to
support his generous work with the modest amount he asks.

The source code for CAR!GOAT is available for study as
CAR-GOAT.ZIP from the GEnie FORTH RoundTable.  Download
file #2398 if you can PKUNZIP; if not, drop me an e-mail
( D.ZETHMAYR1 ) and I'll XMODEM-e-mail it to you.

You don't need to get Pygmy in order to run CAR!GOAT.  It's
language-transparent.  Just start it like any other .COM
or .EXE program you use, such as an editor or ALADDIN.
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