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The two envelopes problem
Posted: 29 July 2012 08:24 AM   [ Ignore ]   [ # 1426 ]
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kkwan - 29 July 2012 07:08 AM
GdB - 26 July 2012 03:36 AM

Exactly the answer to expect: from a difference in description you conclude to a difference in reality.

Not so. (X, 2X) and (X, 1/2X) are distinct possible events, not “a difference in description”.

So I am playing the TEP with you, and I put (10,20) in the envelopes. Is that (X,2X) or (X,X/2).

kkwan - 29 July 2012 07:08 AM

Are your statements actual facts or only merely assertions?

They are meant to be true propositions, and I want you to say if you think them true too or not. But you refuse to answer them, even if they are that simple.

I have two envelopes, one with a green piece of paper, the other with a red one.

1. If I have the envelope with the green one the other contains the red one.
2. If I have the envelope with the red one the other contains the green one.

I have two envelopes with different amounts, call them X and Y:

3. If I have the envelope with X, then the other one contains Y.
4. If I have the envelope with Y, then the other one contains X.

Are above statement true? Yes or no? Why is it so difficult to say ‘yes’? Or why is it so difficult to tell me why they are wrong? Even if these statements were not relevant, you could just say if you think them true or not.

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Posted: 10 August 2012 09:55 PM   [ Ignore ]   [ # 1427 ]
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GdB - 29 July 2012 08:16 AM

Write4U, do you see? He does exactly as I predicted. He introduces dollar cents.

kkwan, suppose I restrict TEP to whole dollar amounts only. So I say to you that the envelopes contain just full dollars. Am I now restricted to even numbers?

Why should the TEP be restricted to whole dollars only?

Are dollars and cents valid amounts of money?

They are.

So, there is no good reason for any restriction wrt whole dollars in the TEP.

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Posted: 10 August 2012 10:35 PM   [ Ignore ]   [ # 1428 ]
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kkwan - 10 August 2012 09:55 PM
GdB - 29 July 2012 08:16 AM

kkwan, suppose I restrict TEP to whole dollar amounts only. So I say to you that the envelopes contain just full dollars. Am I now restricted to even numbers?

Why should the TEP be restricted to whole dollars only?

You should answer a question. It is not ‘why’. The question is if under the condition ‘whole dollars only’ one can still play TEP with all amounts, even and odd.

(And if you have trouble with dollars take dollar cents. But we had this already. You are factually saying that TEP does not work with real bank bills and coins.)

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Posted: 10 August 2012 11:26 PM   [ Ignore ]   [ # 1429 ]
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GdB - 29 July 2012 08:24 AM

So I am playing the TEP with you, and I put (10,20) in the envelopes. Is that (X,2X) or (X,X/2).

The point is, in the TEP, you did not put (10, 20) in the envelopes. So, either (X, 2X) or (X, 1/2X) are in the two envelopes per the TEP.

They are meant to be true propositions, and I want you to say if you think them true too or not. But you refuse to answer them, even if they are that simple.

I have two envelopes, one with a green piece of paper, the other with a red one.

1. If I have the envelope with the green one the other contains the red one.
2. If I have the envelope with the red one the other contains the green one.

I have two envelopes with different amounts, call them X and Y:

3. If I have the envelope with X, then the other one contains Y.
4. If I have the envelope with Y, then the other one contains X.

Are above statement true? Yes or no? Why is it so difficult to say ‘yes’? Or why is it so difficult to tell me why they are wrong? Even if these statements were not relevant, you could just say if you think them true or not.

                           

Obviously, they are true and “that simple” as stated in the two propositions.

However, it is not “that simple” in the context of the TEP as it is possible that either (X, 2X) or (X, 1/2X) are in the two envelopes, but you don’t know what are actually in the two envelopes.

The following reasoning applies:

1. If an envelope is selected and it contains a denoted amount A, either 2A or 1/2A is in the other envelope as you don’t know whether A is the smaller or the larger amount.

2. You don’t know whether (X, 2X) or (X, 1/2X) are actually in the two envelopes.

3. Therefore, you cannot assume (X, 2X) are in the two envelopes as (X, 1/2X) is equally possible.

Thus, your propositions are not relevant (notwithstanding their truth) as they refer to defined explicit situations whereas the TEP is inherently not so.

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Posted: 10 August 2012 11:48 PM   [ Ignore ]   [ # 1430 ]
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kkwan - 10 August 2012 11:26 PM

Obviously, they are true and “that simple” as stated in the two propositions.

Small moves, kkwan, small moves.

Now the following two propositions:

I have two envelopes with amounts X and 2X:

5. If I have the envelope with X, then the other one contains 2X.
6. If I have the envelope with 2X, then the other one contains X.

True or not true? Why (not)?

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Posted: 10 August 2012 11:50 PM   [ Ignore ]   [ # 1431 ]
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GdB - 10 August 2012 10:35 PM

You should answer a question. It is not ‘why’. The question is if under the condition ‘whole dollars only’ one can still play TEP with all amounts, even and odd.

(And if you have trouble with dollars take dollar cents. But we had this already. You are factually saying that TEP does not work with real bank bills and coins.)

It is ridiculous to restrict money to whole dollars, whether odd or even.

There is no problem with dividing any finite amount of money in dollars by 2.

Why must the TEP work with only “real bank bills and coins”?

cheese

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Posted: 11 August 2012 12:10 AM   [ Ignore ]   [ # 1432 ]
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GdB - 10 August 2012 11:48 PM

Now the following two propositions:

I have two envelopes with amounts X and 2X:

5. If I have the envelope with X, then the other one contains 2X.
6. If I have the envelope with 2X, then the other one contains X.

True or not true? Why (not)?

The problem is, in the TEP, you cannot assume that (X, 2X) are in the two envelopes as you don’t know that explicitly because (X, 1/2X) is equally possible.

Thus, you cannot say “I have two envelopes with amounts X and 2X” as a verifiable fact as you don’t know. It is only an assumption.

What you can only say is that “I have two envelopes with either (X, 2X) or (X, 1/2X)”.

As such, if you select an envelope and it contains X, either 2X or 1/2X is in the other envelope.

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Posted: 11 August 2012 12:18 AM   [ Ignore ]   [ # 1433 ]
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kkwan - 10 August 2012 11:50 PM

Why must the TEP work with only “real bank bills and coins”?

Did I say ‘only’? But it should still work with real bank bills and coins: it is the description of TEP. Or how do you read ‘Let us say you are given two indistinguishable envelopes, each of which contains a positive sum of money’?
My slightly rephrased question is ‘Can you play TEP when you use bill and coins?’

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Posted: 11 August 2012 12:21 AM   [ Ignore ]   [ # 1434 ]
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kkwan - 11 August 2012 12:10 AM
GdB - 10 August 2012 11:48 PM

Now the following two propositions:

I have two envelopes with amounts X and 2X:

5. If I have the envelope with X, then the other one contains 2X.
6. If I have the envelope with 2X, then the other one contains X.

True or not true? Why (not)?

The problem is, in the TEP, you cannot assume that (X, 2X) are in the two envelopes as you don’t know that explicitly because (X, 1/2X) is equally possible.

Thus, you cannot say “I have two envelopes with amounts X and 2X” as a verifiable fact as you don’t know. It is only an assumption.

What you can only say is that “I have two envelopes with either (X, 2X) or (X, 1/2X)”.

As such, if you select an envelope and it contains X, either 2X or 1/2X is in the other envelope.

I did not say it is TEP. If you can say that the propositions for X and Y are true, why can’t you if I say X and 2X?

So again, kkwan. You are in ‘avoiding mode’ again. Will you switch to ‘refusing mode’? Or will you tell me if the propositions are correct or not?

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Posted: 11 August 2012 04:14 PM   [ Ignore ]   [ # 1435 ]
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kkwan
In the TEP, it is not so obvious as in your examples. Either (X, 2X) or (X, 1/2X) are in the two envelopes but you don’t know which are actually in the two envelopes.

IMO, that is a false (misleading) statement.  It is true that one might look at the problem both ways in theory, but in reality there are only two amounts, not three.
So no matter what you call the “values”, they remain constant and there is is only one solution which satisfies both X, 2X and (not or) X, 1/2X at the same time.

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Posted: 13 August 2012 07:12 AM   [ Ignore ]   [ # 1436 ]
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GdB - 11 August 2012 12:18 AM

Did I say ‘only’? But it should still work with real bank bills and coins: it is the description of TEP. Or how do you read ‘Let us say you are given two indistinguishable envelopes, each of which contains a positive sum of money’?
My slightly rephrased question is ‘Can you play TEP when you use bill and coins?’

A positive sum of money can be expressed in dollars/cents with currency or as integers/decimals.

So, if a positive sum of money is divisible by 2 (with one cent as the minimum unit of currency), there is no restriction for playing the TEP with bills and coins.

However, it is cumbersome.

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Posted: 13 August 2012 07:35 AM   [ Ignore ]   [ # 1437 ]
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kkwan - 13 August 2012 07:12 AM

So, if a positive sum of money is divisible by 2 (with one cent as the minimum unit of currency), there is no restriction for playing the TEP with bills and coins.

Now following Write4U, there are 15 and 30 dollar cents in the envelopes. What is (X,2X) and (X,X/2)?

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Posted: 13 August 2012 08:11 AM   [ Ignore ]   [ # 1438 ]
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GdB - 11 August 2012 12:21 AM

I did not say it is TEP. If you can say that the propositions for X and Y are true, why can’t you if I say X and 2X?

So again, kkwan. You are in ‘avoiding mode’ again. Will you switch to ‘refusing mode’? Or will you tell me if the propositions are correct or not?

It is true if and only if you know X and 2X are actually in the two envelopes.

However, in the TEP, you don’t know whether (X, 2X) or (X, 1/2X) are actually in the two envelopes.

If you select an envelope, you don’t know:

1. Whether it contains the smaller or the larger amount.

2. Whether (X, 2X) are actually in the two envelopes as (X, 1/2X) is also possible.

So, if you select an envelope and denote the amount in it as A:

1. If A is the smaller amount, the other envelope contains 2A.

2. If A is the larger amount, the other envelope contains 1/2A.

Therefore, either (A, 2A) or (A, 1/2A) are in the two envelopes.

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Posted: 13 August 2012 08:23 AM   [ Ignore ]   [ # 1439 ]
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Write4U - 11 August 2012 04:14 PM

IMO, that is a false (misleading) statement.  It is true that one might look at the problem both ways in theory, but in reality there are only two amounts, not three.
So no matter what you call the “values”, they remain constant and there is is only one solution which satisfies both X, 2X and (not or) X, 1/2X at the same time.

In each the mutually exclusive events (X, 2X) or (X, 1/2X) there are only two amounts.

For any finite value of X, (X, 2X) and (X, 1/2X) are distinct events.

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Posted: 13 August 2012 08:31 AM   [ Ignore ]   [ # 1440 ]
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GdB - 13 August 2012 07:35 AM

Now following Write4U, there are 15 and 30 dollar cents in the envelopes. What is (X,2X) and (X,X/2)?

(X, 2X) is (15, 30) and (X, 1/2X) is (15, 15/2) in dollars/cents.

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