I’ll have a look GdB, but you keep ducking the real issue.
In the TEP opening your envelope makes no difference.
But you are saying yes it does and you should switch once if you open it.
You need to fix this.
No, you do not see through the ‘abstraction layer’.
Yes.
I’m keeping the puzzle precisely the same except that you pick an envelope then look and then decide whether to switch or not.
The reason looking can’t make a difference is because whatever number you have it will be right to switch!
Because of this the same equation needs to work whether you look or don’t.
I think what is blocking you from seeing this is that you think there must be a simple solution.
Examples
nearly-TEP:
You say one envelope contains $10 and the other $20, but not which amount is in which envelope.
Now there are 4 possibilities:
I picked the $10 first, switch, and get $20
I picked the $20 first, switch, and get $10
I picked the $10 first, do not switch, I get $10
I picked the $20 first, do not switch, I get $20
So switching makes no difference.
Now I only know that the one envelope contains an unknown amount, and the other twice as much.
Call the unkown amount A
I picked $A first, switch, and get $2A
I picked $2A first, switch, and get $A
I picked the $A first, do not switch, I get $A
I picked the $2A first, do not switch, I get $2A
So switching makes no difference.
TEP, and opening the first envelope, I see I picked $10
Now there are only 3 situations:
I keep the $10, I get $10
I switch, and get $5
I switch, and get $20.
In the non-TEP situation I am not allowed to choose the box first. That is the difference.
You are allowed to choose the envelope first.
You have to argue that looking makes a difference, which can’t be done.
Or accept the two situations looking or not looking are the same, in which case you need an equation that works in both cases.
I put 2 amounts into 2 envelopes, in such a way that one amount is double of the other one. You can describe this situation as
1. “one envelope contains A and the other 2A”.
You don’t know which amount is in which envelope. But it is just not true that
2. “I put the same A into an envelope and A/2 in the other one”
at the same time.
It does not matter which of the descriptions I take, they are equivalent. But they do not go together. In stating that the other envelope could contain 2A or A/2 you use 2 descriptions that cannot be true at the same time.
Under description 1, I can gain or loose A.
Under description 2, I can gain or loose A/2.
As this is the TEP situation, whatever the description 1 or 2 I take my gain or loss equals out. But I cannot mix description 1 and 2. Important: I do know the amounts, I put them into the envelopes, only you don’t know.
Now I give you $X. I toss the coin, and dependent on that I put 2X or X/2 into the next envelope.
Now I do not have 2 true descriptions, but I do not know which one applies.
So in TEP I have 2 descriptions that both describe the situation correctly, but I do not know which envelope I picked first. So I can pick one of the descriptions, and so I conclude that my possible gain and loss are equal.
In non-TEP I have no idea what the correct description is.
I’m comparing the TEP with the TEP in which you look in the envelope before deciding whether to switch or not. That’s the only difference, everything else remains precisely the same.
Because looking in the envelope can make no difference you need an equation that fits both situations equally.
My remark was about the pure TEP. I can describe TEP as A and 2A or as A/2 or A, but not both at the same time. From there on I would say TEP is solved.
StephenLawrence - 26 January 2012 11:04 AM
I’m comparing the TEP with the TEP in which you look in the envelope before deciding whether to switch or not. That’s the only difference, everything else remains precisely the same.
Because looking in the envelope can make no difference you need an equation that fits both situations equally.
You are right, the problem is a little bit more complicated than I originally thought, but there is a solution. Think for a moment about such propositions as ‘God exists, or he does not exist’, so the chance that he exists is 50%, or ‘today I will get struck by lightning, or not’ so the chance I will survive today is 50%. I suppose you see that this kind of reasoning is wrong. Otherwise I do not have to continue.
So let’s look for the chances of having A or 2A, pure TEP. This means I put in total 3A in the envelopes. Without knowing anything it is still pure TEP. Now I open the envelope, and see the amount. I don’t know if that is A or 2A, but now my chance estimate might change. Suppose I am doing this game with you, I put the amounts in. I put $1 in one envelope and $2 in the other, and we sit in a pub, and we have already been drinking for some time, say for $50.
Now you found there is $2 in the envelope you picked. Could there be $4 in my closed envelope? How big is the chance? Now I add a little information: for some minutes I said ‘Ups, my money ran out, I can only afford me a cup of tea now’. A cup of tea costs $3. So what is the chance that I have put $4 in my envelope? Zero of course, I do not have enough money, so you know there is $1 in it.
In short, as soon as I see the money, without any other indications, I have no idea of the chances. But there is no reason to believe they are 50/50, it depends on the situation.
I’m comparing the TEP with the TEP in which you look in the envelope before deciding whether to switch or not. That’s the only difference, everything else remains precisely the same.
Because looking in the envelope can make no difference you need an equation that fits both situations equally.
Looking into the selected envelope is precisely the same except now A is a known value.
Because A could be either the smaller or the larger amount, there are two possible situations:
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
Since there are only two envelopes and they are identical, the probabilities of 1 and 2 can be considered as equal i.e. as 1/2 and 1/2.
So, the equation that fits both situations is:
In the TEP, the total amount in the two envelopes is neither specified nor known by the player. Thus, it cannot be presumed that it is fixed.
So, (as far as the player can know) in situation 1 the total amount is 3A, whereas in situation 2, the total amount is 11/2A.
Both total amounts satisfy the only known information that “the amount in one envelope is twice that of the other envelope”
This change in the total amounts depending on the situation might seem absurd but, as far as the player can know, it can be so. However, it is conceivable that the progenitor of the TEP could be a computer who decides whether A is the smaller or the larger amount and the situations randomly (as both situations are equally probable), then adjust the total amounts accordingly.
You are right, the problem is a little bit more complicated than I originally thought,
Looks like it, who would have thought it.
but there is a solution.
Sure but what? We need an equation that works in both cases.
Think for a moment about such propositions as ‘God exists, or he does not exist’, so the chance that he exists is 50%, or ‘today I will get struck by lightning, or not’ so the chance I will survive today is 50%. I suppose you see that this kind of reasoning is wrong. Otherwise I do not have to continue.
So let’s look for the chances of having A or 2A, pure TEP. This means I put in total 3A in the envelopes. Without knowing anything it is still pure TEP. Now I open the envelope, and see the amount. I don’t know if that is A or 2A, but now my chance estimate might change. Suppose I am doing this game with you, I put the amounts in. I put $1 in one envelope and $2 in the other, and we sit in a pub, and we have already been drinking for some time, say for $50.
Now you found there is $2 in the envelope you picked. Could there be $4 in my closed envelope? How big is the chance? Now I add a little information: for some minutes I said ‘Ups, my money ran out, I can only afford me a cup of tea now’. A cup of tea costs $3. So what is the chance that I have put $4 in my envelope? Zero of course, I do not have enough money, so you know there is $1 in it.
In short, as soon as I see the money, without any other indications, I have no idea of the chances. But there is no reason to believe they are 50/50, it depends on the situation.
I have absolutely no idea what you are talking about.
I have absolutely no idea what you are talking about.
That’s a pity.
Step by step.
“Today I will get struck by lightning, or not. Conclusion: the chance that I will be struck by lightning is 50%’”
Correct or not?
“This diagnostic test is 90% reliable. This means that when there is 10% chance the test is positive, but I do not have the illness. My test was positive. So 90% chance I have the illness.”
Correct or not?
O, sorry, forget to say that we know that only 1% of the people really have the illness.
I have absolutely no idea what you are talking about.
That’s a pity.
Step by step.
“Today I will get struck by lightning, or not. Conclusion: the chance that I will be struck by lightning is 50%’”
Correct or not?
Incorrect, the mistake is not factoring in that getting struck by lightening is very unlikely.
“This diagnostic test is 90% reliable. This means that when there is 10% chance the test is positive, but I do not have the illness. My test was positive. So 90% chance I have the illness.”
Correct or not?
O, sorry, forget to say that we know that only 1% of the people really have the illness.
Incorrect, the mistake is that if we were to test 100 people 10 of them would be positive and yet they can’t all have a 90 % chance of having the illness because only 1 in 100% has it.
Don’t know the proper sum off the top off my head but can see the mistake.
Trouble is this doesn’t help us with the problem at hand, we know there is a mistake but don’t know what it is.
So I open my envelope and it has £10 in it.
What I know is it might be the smaller amount and it might be the larger amount with a 50/50 chance.
So what I know is the other envelope might contain £5 and might contain £20 with a 50/50 chance
And as it’s double or half with a 50/50 chance I should switch.
But of course really it makes no difference whether I do or not.
Incorrect, the mistake is that if we were to test 100 people 10 of them would be positive and yet they can’t all have a 90 % chance of having the illness because only 1 in 100% has it.
OK. So the chance of being struck by lightning is not 50%, and the chance I have the illness is not 90%, even if the diagnostic test is 90% reliable.
Also think about the quiz master problem. It is reasonable to switch, when I know the history. For somebody just coming in, knowing nothing about the third option, the choice is even.
So context matters. Are you sure that the chance of the other envelope containing A/2 or 2A is the same? You see that in the previous examples these kinds of reasoning are wrong.
So context matters. Are you sure that the chance of the other envelope containing A/2 or 2A is the same? You see that in the previous examples these kinds of reasoning are wrong.
That’s like asking if I’m sure the chance of a coin landing on heads or tails is the same.
Context matters and in this context, yes of course.
