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Archive Magazine 1996
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Gamble02
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1993-05-01
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%OP%VS4.13 (28-Apr-92), Gerald L Fitton, R4000 5966 9904 9938
%OP%DP0
%OP%IRY
%OP%PL0
%OP%HM0
%OP%FM0
%OP%BM0
%OP%LM4
%OP%PT1
%OP%PDPipeLine
%OP%WC1026,2262,188,1748,0,0,0,0
%CO:A,72,72%
%C%Open The Box
%C%by Gerald L Fitton
Keywords:
Gamble Fitton
The Problem
This is the statement of the problem copied from an earlier article.
The solution follows.
I have seen many variants of the following problem. One appeared on
the Februaryá1993 PipeLine disc under the heading of 'conditional
probability' but, as it was stated on that disc, it seems to have been
too difficult for many of you. I've had only incorrect replies. I owe
the earlier version to Robert Macmillan of Colton Software; this
version is courtesy of a lady called Marilyn vos Savant quoted in the
Sunday telegraph on the 7tháFebruaryá1993. Her version has two
advantages over the Robert Macmillan version; the first is that it is
easily testable ÿ the second is that it contains better clues to its
solution. This is a spreadsheet problem. No prizes for the correct
answer but you might become famous. How? I'll mention your name!
You are a game show contestant and, through sheer brilliance, you have
won the right to open one of three boxes. One of the three contains
the Star Prize, the other two contain Booby Prizes. The game show host
knows which box contains the Star Prize but, of course, you don't.
You choose a box and tell the game show host and the world wide
audience of your choice. In accordance with the rules of the box
opening procedure, the host host opens one of the other two boxes which
he knows to contain a Booby Prize! At this stage, the rules allow you
to stick with your original choice or change to the other unopened box.
The question you have to answer is "Would changing to the other
unopened box improve your chances of winning?"
Now, I don't want to hear of your inspired guesses. I don't want a
whole load of conditional probability theory, and I don't want to hear
about Bayes' theorem (or anyone else's for that matter - unless you
want to name one after me!). What you have to do to achieve fame, if
not fortune, is to construct a spreadsheet which simulates many runs of
the 'Open the Box' procedure and use the result to 'prove' your answer
to the question.
Oh yes! A hint. My son once told me that I'm not a 'Computer person'.
His definition of a 'Computer person' is someone who starts counting
(and labelling boxes) with zero instead of one! Label your boxes 0, 1
and 2. Use the statistical function rand() to decide which box
contains the Star Prize, which box you have chosen and which box is
selected by the game show host. If you use one row of the spreadsheet
per 'trial' then you can total about 100 rows to determine the better
strategy.
To swop or not to swop? That is the question.
The Solution
I have received a good spreadsheet solution from Bruce D Brown which
you will find in the directory called Brown.
I did receive another solution. It was a well reasoned argument from
Jonathan Ormond. However, he didn't do it with a spreadsheet so, with
some reluctance, I have nothing much to say about his solution except
that it used the usual 'conditional probability' method without the
usual formula. It was a very good piece of reasoning without any
mathematical jargon.