Re: The Double or One Half Paradox
From: Carl Cotner (cfc-usenet_at_tau.aauetiu.net)
Date: 07/05/04
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Date: Mon, 5 Jul 2004 14:10:54 -0400
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:
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.)
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".
There is more, but it seems silly at this point.
Regards,
Carl
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