h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

This is how it goes:

Say your professor tells you "next week, I will make a surprise pop-quiz, it'll be totally unexpected".

Now, the test can't be in Friday, since if Thursday you go home and the test hasn't occured yet, you know, that the test will take place in Friday. That's not unexpected.

So, having covered this, the test can't be in Thursday neither, since if you go home after Wednesday, and the test hasn't occured yet, and we already know that the test can't take place in Friday, you know, that the test will take place in Thursday. That's not unexpected either.

By the same logic, the teacher can't actually surprise you, any day of the week.

But all of us students know that this is not true, and the pop-quiz is always surprising, no matter which day it is.

Anybody got a solution?

The resolution of the paradox is that there is a small but finite chance that the professor is lying and either there will not be a surprise quiz next week at all, or the surprise quiz will not be totally unexpected. Given that, if there is no quiz Mon, Tue, Wed, Thu, then you would not be able to garantee that there *will* be a quiz.

Example: define 'totally unexpected' in this context as it being impossible to predict whether or not there'll be a quiz with > 50% accuracy.
Set:
P(quiz on monday) = 1/2
P(quiz on tuesday) = 1/4
P(quiz on wednesday) = 1/8
P(quiz on thursday) = 1/16
P(quiz on friday) = 1/32
P(no quiz) = 1/32

The stats resolve the paradox provided you accept a ~3% chance that the professor is lying and the above definition of 'totally unexpected'. There are other similar resolutions. the key point isn't that the professor *is* lying, but the professor *might* be lying - that tiny chance allows the statement to be correct the vast majority of the time.

-Martin

If, through that logic, you've decided that the professor can't spring a test any of the days, then whichever day he decides to do it will be a surprise.

The problem as stated is flawed. The second post to this is technically correct, but it is too verbose


The only day the quiz can happen and not be a surprise is Fri., because the only day you can know for sure there will be a quiz, if it hasn't happened yet, is thurs. But in the high likelihood the quiz is not Fri., then you will be surprised by it. The problem states that you can know on wed. that it won't happen on thurs, but this is not true. But if I asked you on any other day if there was going to be a quiz tomorrow, you wouldn't know, thus making it a surprise.

"Life can be understood backwards, but we will have to live it forwards"- someone says somewhere.

"Life can only be understood backwards, but it must be lived forwards."
-- Soren Kierkegaard

Thus spake Google, anyway. *shrug*

The Prof. is just going to give it any day he feels like it.

He could give a test on Monday *and another one on Tuesday*. The second one would be unexpected, wouldn't it?