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The two envelopes problem
 Posted: 10 January 2012 07:09 AM [ Ignore ]   [ # 46 ]
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StephenLawrence - 09 January 2012 11:04 PM

It’s irrational to take that into account.

That’s the solution.

Why is it irrational?

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 Posted: 10 January 2012 08:41 AM [ Ignore ]   [ # 47 ]
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kkwan - 10 January 2012 07:09 AM
StephenLawrence - 09 January 2012 11:04 PM

It’s irrational to take that into account.

That’s the solution.

Why is it irrational?

Why not?

There are 2 envelopes, envelope 1 with amount A and envelope 2 with amount 2A.
There are 4 possible situations:
a. I took envelope 1 and do not switch: I get A. Gain -A
b. I took envelope 2 and do not switch: I get 2A. Gain: A
c. I took envelope 1 and switch: I get 2A. Gain: A
d. I took envelope 2 and switch: I get A. Gain: -A

So chances of gaining A when switching 2/4 = 1/2 (i.e. 50%)
Chances of losing A: 2/4 = 1/2, also 50%.

Are you just trolling, or do you really believe that you are right?

I assume I can prove to you that 1 equals 2:

Suppose:
a = b who are non-zero
Multiply both sides by a:
a^2= ab
Subtract b^2:
a^2 - b^2 = ab - b^2
Factor:
(a - b)(a + b) = b(a - b)
Now we can take (a - b) away on both sides:
a + b = b
Now we said that a = b:
b + b = b
Put them together:
2b = b
Divide by b:
2 = 1

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 Posted: 10 January 2012 10:50 AM [ Ignore ]   [ # 48 ]
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StephenLawrence - 09 January 2012 11:28 PM

You’re viewing it as similar to this:

I have £10

On the toss of a coin I’m offered double or half so I should do it.

In the this case double or half is two different amounts £5 or £ 10 so I can gain more than I can lose.

Not so wrt the two envelopes.

Initially, you have nothing.

You selected an envelope but you don’t know whether the envelope you selected has £10 or £20 or whether its amount is larger or smaller than the other envelope.

Either:

You selected an envelope with £10 in it. On switching you have £20 in the other envelope

Or:

You selected an envelope with £20 in it.  On switching you have £10 in the other envelope.

You end up with either £20 or £10.

Now, if you selected the £10 envelope and did not switch, you have missed the opportunity to get another £10. (100% gain)

OTOH, if you selected the £20 envelope and did not switch, you have avoided the loss of £10. (50% loss)

You end up with either £10 or £20.

However, the order of £10 and £20 is reversed wrt switching and not switching.

So, why switch if the final outcomes are £20 or £10, £10 or £20?

That might be the actuality for one instance, but what about multiple instances?

Because you don’t know what is in both envelopes and the odds are even, the guide to switch or not to switch is the expected value:

Where A is the amount in the selected envelope.

Whether A is £10 or £20, the expected value is more than A ....... you gain on average by switching.

If there were multiple instances of choosing the two envelopes, you gain on average by switching which means that there will be more £20 verses £10 outcomes and vice versa for not switching.

Paradoxically, this is recommended ad infinitum in steps 9, 10 and 11.

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 Posted: 10 January 2012 11:44 AM [ Ignore ]   [ # 49 ]
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kkwan - 10 January 2012 10:50 AM

Now, if you selected the £10 envelope and did not switch, you have missed the opportunity to get another £10. (100% gain)

100% of £10

OTOH, if you selected the £20 envelope and did not switch, you have avoided the loss of £10. (50% loss)

50%  of £20

You end up with either £10 or £20.

However, the order of £10 and £20 is reversed wrt switching and not switching.

So, why switch if the final outcomes are £20 or £10, £10 or £20?

That might be the actuality for one instance, but what about multiple instances?

Because you don’t know what is in both envelopes and the odds are even, the guide to switch or not to switch is the expected value:

Where A is the amount in the selected envelope.

Whether A is £10 or £20, the expected value is more than A ....... you gain on average by switching.

If there were multiple instances of choosing the two envelopes, you gain on average by switching which means that there will be more £20 verses £10 outcomes and vice versa for not switching.

Paradoxically, this is recommended ad infinitum in steps 9, 10 and 11.

What is happening is your sum is taking acoount of the 100% gain or 50% loss but what is not included in your sum is the cancelling out effect of the fact that the 100% gain is of half the amount and the 50% loss is of double the amount.

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 Posted: 10 January 2012 11:54 AM [ Ignore ]   [ # 50 ]
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It certainly is not irrational.  All of the numbers you are dealing with are rational.

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 Posted: 10 January 2012 12:50 PM [ Ignore ]   [ # 51 ]
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GdB - 10 January 2012 08:41 AM

There are 2 envelopes, envelope 1 with amount A and envelope 2 with amount 2A.
There are 4 possible situations:
a. I took envelope 1 and do not switch: I get A. Gain -A
b. I took envelope 2 and do not switch: I get 2A. Gain: A
c. I took envelope 1 and switch: I get 2A. Gain: A
d. I took envelope 2 and switch: I get A. Gain: -A

So chances of gaining A when switching 2/4 = 1/2 (i.e. 50%)
Chances of losing A: 2/4 = 1/2, also 50%.

Chance of gaining A when switching is 1/4. In b. there is no switching.

In a. and b. if the contents are known and are compared to the amounts in the other envelopes, then gain is -A and A respectively.

But, it is not known which envelope has A or 2A. Randomly selecting one gives you no information that it is either A or 2A.

Thus, by not choosing the option to switch, the loss/gain are not actualized, therefore:

a. Gain/loss = unknown
b. Gain/loss = unknown
c. Gain = A
d. Loss = A

Chance of gain A when switching is 1/4 (25%)
Chance of loss A when switching is 1/4 (25%)
Chance of unknown gain/loss by not switching is 2/4 = 1/2 (50%)

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 Posted: 10 January 2012 03:14 PM [ Ignore ]   [ # 52 ]
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I agree with kkwan (a dubious endorsement, I know)

All this switching does not influence the outcome one bit.
Every equation that has been presented favoring one or the other can also be presented in reverse order.

This is how I see it
a gain of +10 has a certainty of 100%
a gain of +20 has a probabuility of 50% regardless which envelope you chose first or how many times you may switch.
at no time do you actually posess +20, but only a 50% “chance” at +20

Any regrets of ‘not being rewarded’ with +20 and viewing it as a ‘loss’ is a psychological illusion.

It remains a matter of luck of the draw, which only materializes when opening the envelope of choice.

One might even invoke “vagueness”.........

[ Edited: 10 January 2012 03:52 PM by Write4U ]
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 Posted: 10 January 2012 04:12 PM [ Ignore ]   [ # 53 ]
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let me reduce the problem to its most simple form.

Someone places two envelopes on the table in front of you and says. “One envelope contains more money than the other. Take your pick.”

How can you possibly ascertain which envelope has the larger amount? And if you are never told the actual amounts, how would you even know if you picked the envelope with the larger or the lesser amount?

This is an exercise in futility. The introduction of amounts is used only as a false “known”, which causes a psychological expectation of gain or loss.

edited to add: The fallacy lies in the term ‘switching’. In reality you can only pick one or the other.

[ Edited: 10 January 2012 07:37 PM by Write4U ]
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 Posted: 10 January 2012 08:31 PM [ Ignore ]   [ # 54 ]
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What am I? Stephen Hawkings?  Give me an envelope.  That’s my envelope and I’m stickin to it.

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 Posted: 10 January 2012 08:40 PM [ Ignore ]   [ # 55 ]
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TimB - 10 January 2012 08:31 PM

What am I? Stephen Hawkings?  Give me an envelope.  That’s my envelope and I’m stickin to it.

There you go…..

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 Posted: 10 January 2012 09:18 PM [ Ignore ]   [ # 56 ]
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StephenLawrence - 10 January 2012 11:44 AM

100% of £10

Yes, by fixing A= £10 as a known amount, but we don’t know the value of A…....is it the lower or higher amount and what if you switch? You will never know because you selected an envelope randomly and did not switch.

50%  of £20

Again, only if A is a fixed known value but it is not known.

What is happening is your sum is taking acoount of the 100% gain or 50% loss but what is not included in your sum is the cancelling out effect of the fact that the 100% gain is of half the amount and the 50% loss is of double the amount.

What is this “cancelling out effect” and what/how/why does it cancel wrt the expected value? Please explain.

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 Posted: 10 January 2012 09:21 PM [ Ignore ]   [ # 57 ]
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Write4U - 10 January 2012 03:14 PM

One might even invoke “vagueness”.........

And potential…............

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 Posted: 10 January 2012 09:26 PM [ Ignore ]   [ # 58 ]
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TimB - 10 January 2012 08:31 PM

What am I? Stephen Hawkings?  Give me an envelope.  That’s my envelope and I’m stickin to it.

Good strategy, but are you not interested to get double the amount by switching?

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 Posted: 10 January 2012 09:42 PM [ Ignore ]   [ # 59 ]
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Write4U - 10 January 2012 04:12 PM

How can you possibly ascertain which envelope has the larger amount? And if you are never told the actual amounts, how would you even know if you picked the envelope with the larger or the lesser amount?

Exactly, unless you have x-ray vision, some fantastic premonition or you peeked.

This is an exercise in futility. The introduction of amounts is used only as a false “known”, which causes a psychological expectation of gain or loss.

Quite so, but the expected value can be calculated and it is more than A (whatever its value).

The fallacy lies in the term ‘switching’. In reality you can only pick one or the other.

But, you are offered a switch without prejudice after you have selected the envelope.

If a casino offered this game, all who played could walk out richer (whether if they switched or not) and those who are forever switching get richer ad infinitum whereas the casino will go bust.

[ Edited: 10 January 2012 09:52 PM by kkwan ]
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 Posted: 10 January 2012 10:13 PM [ Ignore ]   [ # 60 ]
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kkwan
But, you are offered a switch without prejudice after you have selected the envelope

Yes, you can switch a million times and twenty years, but you don’t get to open the envelope until you have made your FINAL pick, i.e.

TimB - 10 January 2012 08:31 PM
What am I? Stephen Hawkings?  Give me an envelope.  That’s my envelope and I’m stickin to it.

[ Edited: 10 January 2012 10:19 PM by Write4U ]
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