Re: The Double or One Half Paradox

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 07/05/04 

Date: Mon, 05 Jul 2004 14:56:57 -0500

On 5 Jul 2004 12:43:10 -0700, chvol@aol.com (Charlie-Boo) wrote:

>David C. Ullrich <ullrich@math.okstate.edu> wrote
>> On 3 Jul 2004 19:24:40 -0700, chvol@aol.com (Charlie-Boo) wrote:
>>
>> >There are two boxes on a table, one of which contains twice as much
>> >money as the other. You are allowed to take one. You do so, but
>> >before you open it you are allowed to switch boxes. Should you
>> >switch?
>> >
>> >No: The choice was random. The other box is equally random.
>>
>> That's correct.
>>
>> >Yes: If the box you have now contains X then the other box contains
>> >either X/2 or X*2, for an expected value of 1 1/4 * X, which is > X.
>> >
>> >The solutions that I have seen either say that the answer is No and
>> >don't refute the logic for Yes, or say that the question is
>> >meaningless because an unbounded random number with even distribution
>> >has no average value or can't exist.
>>
>> I don't think the question is meaningless for this reason,
>> because there's no reason we need to be talking about
>> unbounded random variables; we may as well assume that
>> one box contains one dollar and the other contains two,
>> without changing the problem.
>
>Interesting variation, but you no longer have the full force of the
>original argument. You can no longer claim that there is always a
>chance of doubling your holding by switching envelopes. Neither
>puzzle is a true paradox, but I think your variation is less
>convincing.

Everything I said applies without change, except in the
notation, if we don't make the assumption that one box
contains one dollar about one contains two. See the
post where I replied to myself.

>Speaking of variations: There is a gambler's convention in town and
>the convention hall is full of tables offering the game (a benefit of
>the convention registration fee.) You decide to try your luck, but
>are only allowed to play once. However, you notice that the rules
>stipulate that the game is over once you open the envelope, and you
>also notice that you can use the old magician's trick of holding the
>envelope to your forehead and peeking through the fold in the flap.
>So you decide to take advantage of this loophole and subterfuge. You
>set your sites on a $100 prize and play at several tables, but
>prematurely leave each in a huff until you get to one in which you
>have exactly $100 (carrying more than $100-200 in a strange city is
>too dangerous.) Upon happening upon such luck, do you bother to
>switch envelopes before opening your ultimate prize?
>
>It is variations like these that would be useful in building a truly
>general model that formally explains each problem.

There already exists a general framework that suffices to
analyze problems like these. It's called probability theory.

>> ************************
>>
>> David C. Ullrich

************************

David C. Ullrich

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