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 14:42:09 -0500

On Mon, 5 Jul 2004 14:10:54 -0400, Carl Cotner
<cfc-usenet@tau.aauetiu.net> wrote:

>On 2004-07-05, David C Ullrich <ullrich@math.okstate.edu> wrote:
>>>
>>>Below is a proof[*] that the first claim you made with respect to this
>>>puzzle
>>>
>>> If you don't know anything about the distribution of the "random"
>>> amounts then clearly [there is no strategy which allows for a better
>>> expected outcome that just choosing one envelope at random]
>>>
>>>is incorrect (the proof does not assume that there exists a
>>>distribution of the "random" amounts).
>>
>> Unless I'm missing something,
>
>Yes, that's correct.
>
>> the proof below depends on looking
>> inside the first box; if the first box contains d dollars you
>> switch with probability f(d). But the statement of the problem
>> _explicitly_ ruled out looking inside the first box:
>>
>> "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?"
>>
>> Note the words "before you open it".
>
>Among other things you are missing the article that started this
>sub-thread, and your response:

Uh, yes, I'd forgotten about that, sorry. Yes, you're right
what I said about that was false, and yes you gave a valid
proof that it was false.

> On Sun, 4 Jul 2004 07:57:51 -0400, Carl Cotner
> <cfc-usenet@tau.aauetiu.net> wrote:
> >
> >Here's a more interesting question:
> >
> >Suppose you are allowed to open one envelope to see how much money it
> >contains before possibly choosing the other. Is there any strategy
> >which allows for a better expected outcome than just choosing one
> >envelope at random?
>
> If you don't know anything about the distribution of the "random"
> amounts then clearly not.
>
>Note the words "Here's a more interesting question:" and "Suppose you
>are allowed to open one envelope to see how much money it contains
>before possibly choosing the other".
>
>My last post was specifically addressing the words "Is there any
>strategy which allows for a better expected outcome than just choosing
>one envelope at random?" and your response "If you don't know anything
>about the distribution of the 'random' amounts then clearly not."
>
>> My claim was that "x is random but follows no particular
>> distribution" is meaningless.
>
>No, that's not correct. Your actual claim was
>
> But things don't just happen. The distribution of the amount
> is determined by whatever method was used to determine what
> the amount should be. (Which includes asking someone to make
> up a number at random - there there's no way to know what the
> distribution _is_, but that's very different from saying
> there's no distribution.)

That's not the claim I was referring to. The claim I was
referring to was... oops, it wasn't stated as a "claim"
at all:

">>>(Note: The problem doesn't assume any particular distribution
>>>on the possible amounts of money in the envelopes; indeed, it
>>>doesn't assume the possible amounts follow any distribution at all.)
>>
>> ??? If the amounts don't follow _some_ distribution then what
>> do we mean when we say that they're "random"?"

That was intended as a rhetorical question; in asking it
my intent was to claim what I falsely claimed I claimed.

>These seem to be quite different claims (although possibly both are
>incorrect). Amoung other things, the first seems to be a statement
>about definitions (semantics), the other about the physical world
>(physics) and perhaps mathematics.
>
>In any case, you are also missing the discussion of the definition of
>the word "random", in which I gave it a standard definition that does
>not depend on the word or concept "distribution".

I didn't miss the definition you quoted from the OED. I
don't see how that definition has any mathematical
content - you said you assume that the word "random"
usually means something like that in a colloquial
description of a problem, but I don't think it
usually does, I think it usually refers to
the notion as in probabilty. (Or to a fuzzy
understanding of that notion.)

>There is more, but it seems silly at this point.
>
>Regards,
>Carl

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

David C. Ullrich

Relevant Pages

  • Re: The Double or One Half Paradox
    ... >> money as the other. ... >Among other things you are missing the article that started this ... >>contains before possibly choosing the other. ... > amounts then clearly not. ...
    (sci.math)
  • Re: The Double or One Half Paradox
    ... Among other things you are missing the article that started this ... >contains before possibly choosing the other. ... about the distribution of the 'random' amounts then clearly not." ...
    (sci.logic)
  • Re: The Double or One Half Paradox
    ... Among other things you are missing the article that started this ... >contains before possibly choosing the other. ... about the distribution of the 'random' amounts then clearly not." ...
    (sci.math)
  • Missing Money Articles & Documents
    ... The Missing Money Articles & Documents ... U.S.'s Missing $Trillions Make Mainstream At Last, ... Mainstream News Coverage ...
    (alt.gathering.rainbow)
  • Re: French war chest may have gone on dancing girls
    ... We piss away our money on missing rifles and mgs and they piss away theirs on ... It was maintained during the cold war to finance a French ... French war chest may have gone on 'dancing girls' ...
    (soc.retirement)
Loading