Date: Tue, 10 Mar 1998 14:29:00 +0100
From: Hugh Rodwell <m-14970-AT-mailbox.swipnet.se>
Subject: M-TH: Now you see it...


The argument goes:

>	At this point I was a little confused and the problem was
>re-presented to me in terms of a "switch or stick"  strategy.  If you
>always switch to that one remaining curtain, the only way you will lose
>the game is if you got the right curtain the first time.  The probability
>that the "switch" strategy will fail is exactly the same as the
>probability you *shouldn't* switch - the probability you were right on the
>first guess.

In other words you will lose by switching choices if -- and only if -- your
first choice was right. OK? But were you right first time? The chances were
one in three, that is 33%. So if -- and only if -- your first choice was
right, a one in three chance, you'll lose by switching, otherwise you'll
win by switching, and that otherwise is the crunch, since the otherwise of
33% is 66%.

Further exemplification:

>	That only helped slightly, so I considered a case with a thousand
>curtains.  I pick one - there is a miniscule chance that my curtain was
>the right one.  Monty then opens 998 of the thousand curtains, none of
>which has the car behind it.  Now all that remains is the first curtain I
>picked, which is almost certainly the wrong curtain, and the one which -
>998 curtain-opening later - is almost certainly the right one. In that
>case, it seems clear that the car is behind that last, lonely curtain that
>Monty didn't open (99.9% of the time, at least).

Again, you will lose by switching choices if and only if you were right
first time. The chances of being right first time are now one in a
thousand. So you have the choice of continuing with your one in a thousand
choice or switching to the curtain for which the odds have now narrowed
down to 998 in a thousand.

This appears counter-intuitive at first. Maybe Body should demonstrate it
empirically using a De Soto 57 or an Edsel for those listers near enough to
be present (New Jersey isn't it -- Connie Francis country) and stick their
fingers in the holes. Or Rob could reformulate it as a conundrum in
communications and get his students bless their socks to rig up ten
curtains (shoe boxes, whatever) and hide a jar of Vegemite behind one. Then
run it fifty times or so and see if a pattern emerges. Aristotle would have
done it, and if he could, we can.

I don't know what Yoshie could use in Columbus -- shiny boots of leather or
chrome-plated manacles?? Perhaps a *switch*!

Doug  could have copies of Wall Street in the dud booths (sort of
consolation prize) and the equivalent in greenbacks in the right one.

Enjoy.

Hugh





>	The fact that Monty only opens one curtain in the three curtain
>case seems to make us think that we are left with an uninfluenced choice
>between the two remaining curtains.  We tend to isolate that choice from
>its logical context.  We can't see how it is that we can know something
>probabilistically about a curtain that seem to be a perfect unknown.  Once
>you know the trick to the problem it becomes obvious, but the fascinating
>thing to me is how it fools you.  It shows how susceptible we are to
>isolating and, in a sense, fetishizing a logical problem out of context
>and separating it from its premise.
>
>
>
>
>	Self-negation paradoxes are, to my mind, examples of the same
>susceptibility.  In the liar paradox, we know the sentence is a discrete
>entity - a negation - but we feel that we can consider whether the
>sentence can be false about itself separately.  You can't consider the
>sentence from the point of view of itself.  It is a declaration, not a
>riddle.  Next, the "This sentence is non-existent." example reminds us
>that there are things a self-referential sentence simply cannot say.
>Before we consider whether or not it is a false case, we have to consider
>whether or not it is a moot case - an impossibility.  The liar paradoxes
>are not false cases, but moot cases.  "This sentence is true." is an
>equally moot case to "This sentence is non-existent.".  A sentence cannot
>declare itself true, since truth demands comparison.  The context of a
>single sentence is a limiting context.  There are things that cannot be
>said by a single sentence.  You can create two sentence statements that
>are moot:  "The next sentence is true.  The previous sentence was false."
>It has no meaning.
>
>
>
>
>
>
>	peace
>
>
>
>
>
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