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Return to the list of my pages written in English about the two envelopes problem
2021/04/07 12:11:30
First edition 2017/07/25

Two English language Wikipedia articles on the two envelope paradox

This title was revised on November 21, 2017, May 5, 2018.
On April 6, 2021, there are two English language Wikipedia articles on the two envelope paradox.
The older article is hidden behind the redirection. (← Confirmed on April 7, 2021)

Caution
I who am a Japanese wrote this page in English, but I am not so good at English.

How to read the older article "Envelope paradox" of the English language Wikipedia

FirstOpen a page of the English language Wikipedia.
Second  Enter "Envelope paradox" as the search key word, and click the search button.
ThirdIf the article "Two envelopes problem" is shown, click the link on the line "(Redirected from Envelope paradox)".
FourthIf the article "Envelope paradox" is shown click the link "View history".

Comparison of the two articles

(Revised on July 27, 2017.)

Major difference

To my eyes the difference of the article "Envelope paradox" and the article "Two envelopes problem" is as follows.

Features of the article "Envelope paradox"

(Revised on May 12, 2018.)

In my perception this article had the following features at almost revisions. The following Wikipedia articles have same style.

Features of the article "Two envelopes problem"

(Revised on May 12, 2018.)

In my perception this article had the following features at almost revisions. In 2017, I found the following features at the revisions after late 2014.
(Added on January 27, 2019)
Supprement : A reference that seems to be inappropriate (← Added on February 17, 2019)
The explanation of "E(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2)" was accompanied by reference to an article which presented the following diagnostic methods.
Let random variables X and L denote the chosen amount and the lesser amount, respectively.
Then
E(X| (X/2) is the lesser amount) ≠ E(X| 2X is the greater amount) and
E(L| L is the lesser amount) = E(L| 2L is the greater amount).
As a result of this diagnosis, the expectation formula "E=(1/2)(X/2) + (1/2)2X)" is wrong and the expectation formula "E=(1/2)L + (1/2)2L" is correct.
However, I think that it is inappropriate to refer this method, because the essence of this method differ from the essence of the formula "E(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2)".
The former essence is "giving rationale to other formula", and the latter essence is "correction of the formula". (← Added on March 24, 2019)

How the article "Envelope paradox" had been hidden

(Added on July 26, 2017. Revised on July 27,2017.)

On July 3, 2009, the last redirection from "Envelope paradox" to "Two envelopes problem" has been done after several redirection and canceling redirection.
(↑ Revised on October 4,,2017.)
And after it no edit warring had broken out about the redirection.
Since then usual reader may not notice the existence of the article "Envelope paradox".

My hope

(Revised on July 27, 2017.)

I think that these two articles are creatures of distinct dimensions.
"Envelope paradox" is a creature in the mathematical dimension.
"Two envelopes problem" is a creature in the philosophical dimension.
And I know that at least four editors of the article "Envelope paradox" wanted the survival of it in 2006. (← Added on October 4, 2017.)
So I hope that some editor cancels the redirection from "Envelope paradox" to "Two envelopes problem" and makes the two articles coexist.

Appendix : Transition of the presentation of the standard resolution in the article "Two envelopes problem" (Minor changes are omitted)

This appendix was added on November 24, 2017. Revised on January 14, 2018, January 16-18, 2018, February 12, 2018, May 17, 2018, May 22, 2018.

From the first revision 22:36, 25 August 2005 to the revision 21:47, 3 October 2005

No resolution

section contents
opened version problem with no solution

Revision 22:05, 3 October 2005

Both resolutions

section contents
Lead section closed version problem
Proposed Solution 1 DivideThreeByTwoian's resolution
with inconsistent variable theory
A Second Paradox opened version problem
Proposed Solution 2 mathematically standard resolution
The mathematically standard resolution is exprressed with the following phrase.
The subjective probability changes when we get new information, so our assessment of the probability that A is the smaller and larger sum changes

Revision 20:51, 9 October 2005

(Added on January 15, 2018.)
The order of the problem wording changed.

section contents
Lead section closed version problem
(More suitable for the DivideThreeByTwoian's resolution)
Proposed Solution DivideThreeByTwoian's resolution
with inconsistent variable theory
A Harder Problem opened version problem
Proposed Solution mathematically standard resolution

Revision 01:47, 17 March 2008

Both resolutions

section contents
The problem closed version problem
The problem / Proposed solution DivideThreeByTwoian's resolution
with inconsistent variable theory
A harder problem opened version problem
A harder problem / Proposed solution mathematically standard resolution
The mathematically standard resolution is exprressed with the following phrase.
The subjective probability changes when we get new information, so our assessment of the probability that A is the smaller and larger sum changes.

Revision 18:42, 8 October 2008

Standard resolution deleted

section contents
The problem closed version problem
The problem / Proposed solution DivideThreeByTwoian's resolution
with inconsistent variable theory
  mathematically standard resolution deleted!!!

Surprisingly the mathematically standard resolution was not presented about twenty months.

Revision 20:17, 26 May 2010

DivideTreeByTwoian's resolution deleted
Standard resolution revived on the closed version problem

section contents
The problem closed version problem
  DivideThreeByTwoian's resolution deleted!!!
The problem / Solution mathematically standard resolution?
Or theory of assumption of probability distribution?
on the closed version problem
The mathematically standard resolution is exprressed with the following phrase.
Applying the same logic with different ratios shows that we are implicitly assuming that every positive amount is equally likely - however since this implies a uniform probability over an infinite set of values, it is not a valid prior distribution.

Revision 22:01, 20 December 2010

No resolution

section contents
The problem closed version problem
The problem / Solution No solution presented

Revision 03:12, 1 January 2011

DivideThreeByTwoian's resolution revived

section contents
The problem closed version problem
The problem / Solution DivideThreeByTwoian's resolution
with inconsistent variable theory

Revision 23:46, 3 May 2011

DivideThreeBytwoian's resolution deleted
Standard resolution revived on the closed version problem

section contents
The problem closed version problem
  DivideThreeByTwoian's resolution deleted!!!
Informal solution
Formal solution
mathematically standard resolution
on the closed version problem
The mathematically standard resolution is exprressed with the following phrase.
To be specific, it could never be the case that the other envelope is equally likely to be double or half the first envelope, whatever the amount in the first envelope.

Revision 23:30, 15 August 2011

(Added on February 17, 2018.)
The section "External links" was replaced to a link to the page "Two envelopes problem/sources" which contains these external links.
As a result it became harder to find the links to the following important articles.

Revision 01:31, 8 November 2011

DivideThreeBytwoian's resolution revived

section contents
Problem closed version problem
First resolution DivideThreeByTwoian's resolution
with inconsistent variable theory
Second resolution mathematically standard resolution
The mathematically standard resolution is exprressed with the following phrase.
… then it is simply impossible that whatever the amount A=a in the first envelope, it is equally likely that the second contains a/2 or 2a.

In my perception many big editing has been done in 2011.

Revision 18:44, 8 February 2012

The mathematical standard resolution as an uncommon resolution

section contents
The problem closed version problem
The problem/A common resolution DivideThreeByTwoian's resolution
with inconsistent variable theory
Alternative interpretation/Resolution of alternative interpretation mathematically standard resolution
(Probably not theory of assumption of probability distribution) (← Revised on January 27, 2019)
The mathematically standard resolution is exprressed with the following phrase.
… then it is impossible that whatever the amount A=a in the first envelope might be, it would be equally likely, according to these prior beliefs, that the second contains a/2 or 2a.

Revision 10:47, 18 March 2012

Mathematical standard resolution categorised "Bayesian"

section contents
The problem closed version problem
A common resolution DivideThreeByTwoian's resolution
with inconsistent variable theory
Introduction to resolutions based on Bayesian probability theory/Proposed resolutions mathematically standard resolution
(Probably not theory of assumption of probability distribution) (← Revised on January 27, 2019)

In my perception many big editing about the mathematically standard resolution has been done after 2012.

Revision 03:15, 10 October 2012

(Added on February 17, 2018.)
The link to the page "Talk:Two envelopes problem/Literature" was deleted from the section "See also".
As a result it became harder for readers to find important literatures.

Revision 09:49, 17 November 2014

(Added on November 25, 2017. Revised on November 27, 2017.)
The mathematically standard resolution explained using geometric progression

section contents
Lead section closed version problem
(as a summary of the problem)
Problem closed version problem
Logical resolutions DivideThreeByTwoian's resolution
with inconsistent variable theory
E(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2) as the correct expectation formula
(↑ Revised on June 26, 2018)
(LesserOrGreaterMeanValuean's resolution)
"E=(1/2)2x + (1/2)x" is a simple form of the above..
Mathematical resolutions/Simple mathematical resolution mathematically standard resolution?
Or theory of assumption of probability distribution?

(explained using geometric progression)
The explanation of the mathematically standard resolution has many sentences so it is not easy-to-understand for me.
(↑ Revised on May 15, 2018)

Especially, there is a sentence which is confusing with the theory of assumption of probability distribution. When I write it in my word, it will be the following sentence.
In order that the probability is always 1/2 whatever amount in the chosen envelope, we apparently believe in advance that all the amounts in the geometric progression equally likely to be the smaller amount.
(↑ Added on January 27, 2019)

Revision 16:15, 22 November 2014

(On January 5, 2020, the revision date was revised)
The statements about the inconsistent variable theory was replaced by other statements regarding the relationship between the smaller amount average and the larger amount average.
(↑ Revised on January 5, 2020)


section contents
Lead section closed version problem
(as a summary of the problem)
Problem closed version problem
Logical resolutions E(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2) as the correct expectation formula
(↑ Revised on June 26, 2018)
(LesserOrGreaterMeanValuean's resolution)
"E=(1/2)2x + (1/2)x" is a simple form of the above..
Mathematical resolutions/Simple mathematical resolution mathematically standard resolution?
Or theory of assumption of probability distribution?

(explained using geometric progression)

Revision 16:46, 12 April 2016

(Revised on November 25, 2017, November 27, 2017.)
Pure DivideThreeByTwoian's opinion presented

section contents
Lead section closed version problem
(as a summary of the problem)
Introduction/Problem closed version problem
Simple resolution "E=(1/2)2x + (1/2)x" is the correct expectation formula.
(pure DivideThreeByTwoian's opinion)
Simple resolutions E(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2) as the correct expectation formula
(↑ Revised on June 26, 2018)
(LesserOrGreaterMeanValuean's resolution)
Bayesian resolutions/Simple form of Bayesian resolution mathematically standard resolution?
Or theory of assumption of probability distribution?

(explained using geometric progression)

Revision 03:24, 21 June 2016

(Added on January 16, 2018)
The following wording was added in the lead section.
… because you stand to gain twice as much money if you switch while risking only a loss of half of what you currently have, …

section contents
Lead section closed version problem
(as a summary of the problem)
Introduction/Problem closed version problem
Simple resolution "E=(1/2)2x + (1/2)x" is the correct expectation formula.
(pure DivideThreeByTwoian's opinion)
Other simple resolutions E(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2) as the correct expectation formula
(↑ Revised on June 26, 2018)
(LesserOrGreaterMeanValuean's resolution)
Bayesian resolutions/Simple form of Bayesian resolution mathematically standard resolution?
Or theory of assumption of probability distribution?

(explained using geometric progression)

Revision 20:02, 15 October 2016

(Added on February 16, 2018)
The not-three-amounts theory was added to the beginning pf the section "Other simple resolutions".

section contents
Lead section closed version problem
(as a summary of the problem)
Introduction/Problem closed version problem
Simple resolution "E=(1/2)2x + (1/2)x" is the correct expectation formula.
(pure DivideThreeByTwoian's opinion)
Other simple resolutions not-three-amounts theory
(in just one sentence "The step 7 assumes that the second choice is independent of the first choice. ")
E(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2) as the correct expectation formula
(↑ Revised on June 26, 2018)
(LesserOrGreaterMeanValuean's resolution)
Bayesian resolutions/Simple form of Bayesian resolution mathematically standard resolution?
Or theory of assumption of probability distribution?

(explained using geometric progression)

Revision 14:55, 28 April 2017

(Added on June 26, 2018)
The following wording was added in the section "Solutions".
… any apparent paradox is generally due to treating what is actually a conditional probability as an unconditional probability …
I think this sentence conforms to the mathematically standard resolution.

section contents
Lead section closed version problem
(as a summary of the problem)
Introduction/Problem closed version problem
Simple resolution "E=(1/2)2x + (1/2)x" is the correct expectation formula.
(pure DivideThreeByTwoian's opinion)
Other simple resolutions not-three-amounts theory
(in just one sentence "The step 7 assumes that the second choice is independent of the first choice. ")
E(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2) as the correct expectation formula
(↑ Revised on June 26, 2018)
(LesserOrGreaterMeanValuean's resolution)
Bayesian resolutions/Simple form of Bayesian resolution mathematically standard resolution?
Or theory of assumption of probability distribution?

(explained using geometric progression)

Revision 02:15, 3 November 2017

(Revised on November 25, 2017, November 27, 2017.)
section contents
Lead section closed version problem
(as a summary of the problem)
Introduction/Problem closed version problem
Simple resolution "E=(1/2)2x + (1/2)x" is the correct expectation formula.
(pure DivideThreeByTwoian's opinion)
Other simple resolutions not-three-amounts theory
(in just one sentence "The step 7 assumes that the second choice is independent of the first choice. ")
E(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2) as the correct expectation formula
(↑ Revised on June 26, 2018)
(LesserOrGreaterMeanValuean's resolution)
Bayesian resolutions/Simple form of Bayesian resolution mathematically standard resolution?
Or theory of assumption of probability distribution?

(explained using geometric progression)

State of the revision 16:51, 12 December 2018

(Added on January 13, 2018. Revised on March 15, 2020)
section contents
Lead section closed version problem
(as a summary of the problem)
Introduction/Problem closed version problem
Simple resolution Equality of the gain and loss by exchange
(Added on March 15, 2020)
"E=(1/2)2x + (1/2)x" is the correct expectation formula.
The not-three-amounts theory is explained in the paragraph "In simple words", using the following findings.
  • Secretly mixing up of two different circumstances
  • Defference from Ali-Baba version
(↑ Revised on March 15, 2020)
Other simple resolutions The not-three-amounts theory is explained with the sentence "The step 7 assumes that the second choice is independent of the first choice. "
E(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2) as the correct expectation formula
(LesserOrGreaterMeanValuean's resolution)
Bayesian resolutions/Simple form of Bayesian resolution mathematically standard resolution?
Or theory of assumption of probability distribution?

(explained using geometric progression)

State of the latest revision 19:48, 6 March 2020

(Added on March 15, 2020)
The "simple solution" section was given a new format, but the content remained the same.

State of the revision as of 17:49, 17 September 2020

(Added on April 7, 2021)
The paragraph beginning with "The step 7 assumes that the second choice is independent of the first choice" was removed from the section "Other simple resolutions".
section contents
Lead section closed version problem
(as a summary of the problem)
Introduction/Problem closed version problem
Simple resolution Equality of the gain and loss by exchange
"E=(1/2)2x + (1/2)x" is the correct expectation formula.
The not-three-amounts theory is explained using the following findings.
Other simple resolutions E(B)= E(2A|A<B)(1/2) + E((1/2)A|A>B)(1/2) as the correct expectation formula
(LesserOrGreaterMeanValuean's resolution)
Bayesian resolutions/Simple form of Bayesian resolution mathematically standard resolution?
Or theory of assumption of probability distribution?

(explained using geometric progression)


Terms



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