X-Message-Number: 7059
From: Peter Merel <>
Subject: The paradox of unexpected goats.
Date: Wed, 23 Oct 1996 23:32:09 +1000 (EST)

Art Quaife writes,

>But this is WRONG. Your initial guess of door #1 had 1/3 chance
>of winning. Monty's opening another door to reveal a goat, as he
>previously promised to do, does not change that probability. But
>since Monty has now eliminated door #3 as a possibility, door #2
>must now have a 2/3 chance of being correct. You should switch,
>and by doing so you *double* your chance of winning the car.

Sorry, probability theory doesn't work like this; an elementary property
of probability theory is that past results do nothing to fix the
odds of new measurements. If I toss a coin and get a head, that tells me
nothing about a new toss. The probability of the new coin toss, and the
probability of the contestant's second pick, must be calculated based
purely on the immediate circumstances of that event, not on its history.
So the odds of the second pick are indeed 50/50.

I'm not, however, attempting to claim Art's prize - so far as I can see,
Art was plainly offering a prize for the wrong answer, and Hara, having
successfully picked the wrong answer, should receive that prize. However
I will offer a more interesting conundrum along similar lines, known as
"the paradox of unexpected execution".

The usual form of the paradox is: say there is a prisoner on death row,
and he's told that he will definitely be executed before the week is out
- before Sunday - but that the day will not be told to him in advance -
it will be unexpected. The prisoner concludes from this that he can't be
executed at all: "Plainly I can't be executed on Saturday, as, if Friday
went by, I must expect to be executed on the last available day -
Saturday.  But this means I can't be executed on Friday, as, knowing
that Saturday is ruled out, Friday would be the last available day. But
then ...". Using induction to rule out the other days, the prisoner
concludes that he can't be executed at all.

Of course, when they come for him on the Wednesday, it is quite unexpected.

I can't offer a prize for putting your finger on the flaw in the
prisoner's reasoning, but I should point out an extensive treatment at
"http://www.ios.com/~leonf/paradox/paradox.html". The nature of the flaw
is the subject of some controversy, and is thought to have a bearing on
Von Neumann's famous "prisoner's dilemma".

For my money, the flaw is a case of assuming the antecedent: the
prisoner can't really rule out Saturday based on the information he has
- all he can really say, should Saturday turn up, is that he's been told
something paradoxical.  The rest of his proof is therefore founded on
his assumption of the conclusion of his proof.

But I wonder how Art's version of probability theory would deal with it?

Peter Merel.


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