[extropy-chat] monty hall paradox again

[David]

David wrote:

> Spike wrote:
>
>>> Could it really be that you don't realise? Could it be that you
>>> do realise and some others don't and that you enjoy that?
>>
>>
>>
>> Oh I enjoy it, but no I do not realize. Read on.
>>
>>>> For that reason, there really is no reason to bother opening
>>>> and looking in the envelope you chose first.
>>>
>>>
>>> Re-read what you wrote when you first outlined the hypothetical
>>> at the top. It is only AFTER an envelope is chosen and opened
>>> revealing that it contains 10 dollars that the messenger offers a
>>> second trade. The messenger did not say from the outset that
>>> whatever the first envelope contained the contents could be
>>> exchanged after being determining for the contents of the other
>>> envelope prior to their being determined. If that whats you
>>> wanted the messenger to say thats a different hypothetical.
>>>
>>> How long has this list been discussing a twisted hypothetical? Brett
>>
>>
>> Paatsch
>>
>> You may be onto something Brett. What difference does it make
>> if the messenger tells you up front that after making a
>> choice, you will be given the option of trading? The
>> paradox doesn't seem to disappear if it is carefully
>> explained up front, but do suggest a modification of
>> the scenario if you can fix it.
>>
>> spike
>>
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>>
> Lets make things worse :)
> If you open the envelope to find 10 zorgs there are two possible
> scenarios :
> 5:10 and you have the larger amount, and
> 10 :20 and you have the smaller.
> Each of these has a 0.5 probability of being the scenario you are in.
> Therefore there are four possible values to the envelopes :
>
> 5
> 10 (high)
> 10 (low)
> 20
>
> giving an average value to an envelope of 11.25 zorgs.
> At least that?s closer to 10 than 12.5 is. :)
>
>
>
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>
Call the larger envelope value LV,
the smaller envelope SV,
the minimum possible amount of zorgs is z.
and i is some integer.

SV must be i * z.
But LV = 2 * SV , therefore SV can only increment in steps of 2z.

This means that for any given range there are twice as many
possible SV values as there are LV values.
This means that the chance that the other envelope is the
smaller one is twice the chance that it is the larger one.

expected value becomes 2/3 * 5 + 1/3 * 20 = 10.