[helene_t]

hrothgar, on Oct 26 2007, 02:43 PM, said:

You are solving for the probability density function that minimizes the chance that a student can accurately guess which day the test will occur.

Uhm ... yes, something like that. The density function that minimizes the expected value of p.cond(testday=x), where x is the day on which the test actually takes place, and by p.cond I mean the probabilities assigned the evening before. (This corresponds to your phrasing if "accurately guess" means flipping a weighted coin and have it right. If it means "getting prepared for the test as soon as the probability exceeds 50%", it's a different matter).

Alternatively, one could minimize the expected value of the "information-theoretical" object function log(p.cond(testday=x)) but that leads to meaningless results, not sure why.

[whereagles]

This is beginning to sound like Godel's theorem... oh well.

[helene_t]

Interesting idea, Whereagles. A probabilistic version of Godel's theorem. Let p[i|j] be the probability that theorem i is true, as calculated by formular j......

[jtfanclub]

gwnn, on Oct 25 2007, 06:46 AM, said:

The professor says to his students: "next week we will certainly have an exam. it can be on any weekday, but nobody will know in advance on which."

Now students start wondering...

"Friday is obviously impossible, because Thursday night it will become the only possibility, so all of us will know"

...

"Thursday is impossible, because Friday's impossible and thus on Wednesday we'd know"

et cetera

"We can't have a test! WTF"

Nah, it's easy.

On Monday, the students take a practice exam.
On Tuesday, there is a 20% chance that the teacher gives them the real exam.
On Wednesday, if they didn't take the test on Tuesday, there is a 25% chance that the teacher gives them the real exam.
On Thursday, if they didn't take the text on Tuesday or Wednesday, there is a 33% chance that the teacher gives them the real exam.
On Friday, if they didn't take the test on Tuesday, Wednesday, or Thursday, there is a 50% chance that they take the real exam, and a 50% chance that the professor declares that the 'practice' exam they took on Monday was in fact the real exam, and here are their grades.

That should maximize surprise and guarantee that nobody will know in advance on which day the real exam will be given.

[helene_t]

Wow jtf, this problem has baffled philosophers for hundreds of years and you just solved it! I never saw that solution before, it's as beautiful as it's simple!

[helene_t]

Hey, I can use a smiley to indicate "this post is a sarcasm", how can I indicate "this post is not a sarcasm"?

But seriously, jtfanclub's solution is, unfortunately, not the "expert attention" they are looking for on wiki. if only because their scenario is a convict who will be executed on a surprise day:

"Sir, it turns out that the scam execution last monday was in fact a real execution, iow you are already dead". More apt for a Monty Python blog than for wiki.

My solution is not what they are looking for either since they are addressing the logical aspect only, not the information-theoretical aspect. But as for the logical aspect, I can't come up with more than "wtp?". The professor's statement is simply false, not more interesting than "I am an elephant".

But I do think wiki should link it to the problem of non-blinded serial-recruitement trial design. That is at least a variant that has a practical application and has been studied seriously.

[EricK]

It becomes clearer if you restrict the problem to just one day:
Professor: "I will give you a surprise test tomorrow"
Now the student can either believe this statement or not. But if he tries to believe it he arrives at a contradiction - i.e. he believes that there will be a test tomorrow and he believes that he will be surprised by there being a test, which contradicts his believing that there will be a test.

So he is forced to disbelieve the statement.

But since the professor's statement was a compound one, there are two ways to disbelieve it - he can disbelieve that there will be a test, or he can disbelieve that he will be surprised. And the student has no rational way of deciding which way to disbelieve the professor's statement.

So, test or no test, whatever happens the next day, the student will be surprised in some way or another - although it seems to me that he will be less surprised in either scenario than if the professor hadn't made a statement at all.

[helene_t]

OK, he will be surprised by the fact that he won't be surprised. Now I see the link to "this statement is false"

[Cascade]

Martin Gardner discussed this paradox in his book "The Unexpected Hanging". As the title suggests his context is the criminal who receives a death sentence but is told he will not know which day of the following week he will be hanged. He reasons as you did that the sentence cannot be carried out. Nevertheless when the hangman arrives on Tuesday (or whenever) it is completely unexpected fulfilling the judges sentence.

[gwnn]

oh no this is horrible, i've read a little about this, it was just 5 minutes, but all of a sudden i don't know anything, i don't know if this keyboard exists, if information is contradictory, or whether or not the purpose of meaning is a rational number. aaahhh

[P_Marlowe]

Echognome, on Oct 25 2007, 04:55 PM, said:

I've seen this paradox before. I'm sure it's discussed plenty on wiki.

I much prefer this simpler one. The statement "I am lying" can only be true if it's false.

Is it really the same?

Your example seems to be a varianton of
Russels Paradox:
http://en.wikipedia.org/wiki/Russell's_paradox

I am not sure, if the original problems is also
just a variation.

With kind regards
Marlowe

[helene_t]

Fluffy, on Oct 25 2007, 03:28 PM, said:

you can also add that the test migth be at any hour unknown, them, over a continous range it won't work I think.

Correct.

[jtfanclub]

helene_t, on Oct 27 2007, 02:16 AM, said:

But seriously, jtfanclub's solution is, unfortunately, not the "expert attention" they are looking for on wiki. if only because their scenario is a convict who will be executed on a surprise day:

"Sir, it turns out that the scam execution last monday was in fact a real execution, iow you are already dead". More apt for a Monty Python blog than for wiki.

You have to have an unkown state. Put the prisoner in a box, with no light or other way of telling the date or time. Then sometime later, take him out of the box, and hang him. Not only will he be surprised on what day it is, you don't even have to tell him what day it is, if you want to change the problem slightly.

Lots of paradoxes can be solved by Shrodinger's Cat. In the hangman's dilemma, you can make the day of the week an unknown state. In the test-taking, you can make whether you took the test an unknown state. Or you can make the truthfulness of the original statement unknown, etc.

I don't know that any of this is really a paradox. It's more like a tautology- in a completely known state, nothing is a surprise.

[helene_t]

Schrödinger's cat, lol. Let's bring the students in a superposition of a test-state and a no-test-state