Friday, February 17, 2012

Friday Paradox:

The Surprise Exam Paradox

Imagine that a professor provides the following blurb on her syllabus:

"In addition to the final paper and final exam, we will have one pop quiz (for 99% of your grade) on some class day between now and the end of the semester. (The topic will also be a surprise.) I won't tell you which day I am going to give the exam, but I will tell you this: I will definitely give an exam on one of the remaining class days, and on that day you will have no good reason to believe that it will be on that day, rather than some other. (This is just what it means for it be a surprise exam, of course.)"

You might object that what the professor has described is impossible. Consider how a sharp student might reply to the prof.

"Well, you can't give the exam on the last day of class, since then we would know that, there being no more classes remaining, you had to give the exam that day; and in that case it would not be a surprise. So we can safely eliminate the last day of class from the list of possible days on which you can give the exam. But then you can’t give it the second-to-last day of class either; for on that day we would know that you couldn’t wait until the last day-since then it would not be a surprise-and so we would know that you had to give it that day. But then, of course, it would not be a surprise. So we can also safely eliminate the second-to-last day of the semester as a possible date for the exam. But then you can't give it the third to last day either… (and so on, until all the remaining dates on which the professor could give such an exam are eliminated.) Therefore, you can't actually give a surprise exam!"

So here’s the paradox: Something is wrong with the student's reply. A professor can clearly give a surprise exam. What went wrong?

(Thanks to Jeff Speaks at Notre Dame for providing materials on paradoxes online. I've borrowed very heavily from this document.)


S Vetterly said...

I feel that giving a quiz on the last day of class could still be considered a surprise quiz. This is because every day up until the last day, the students would not be able to correctly determine whether they would be getting the quiz on that day or the following one.

This is a classic case of analyzing a case of knowledge by various perspectives. By the student's point of view, the only knowledge (s)he has is that the surprise quiz was not given on the days prior to the current one, thus narrowing down the number of days still remaining for the teacher to choose. However, by the third-person omniscient point of view, the shart student is correct. This is because the student begins by assuming that the teacher has waited until the very end of classes to give the quiz. Therefore, (s)he has already eliminated all the class days on which the teacher could have given the quiz between the current day and the last day of class.

This example is similar to the one presented in Michael Huemer's book entitled "Skepticism and the Veil of Perception," in which an English teacher asks her class a question about a book they had just finished reading. In the book, a fisherman finds a pearl that he feels will bring his family wealth and security in the future. However, between the time at which he finds the pearl and that at which he sells it, he experiences many hardships surrounding the object, such as people trying to steal it away from him. The teacher asks whether it the fisherman should have thrown the pearl back rather than keeping it. Most of the students answered yes, and were surprised to find they were wrong. This is because they were approaching the situation from a third-person omniscient point of view. They already knew the hardships that the pearl had brought to the man, and felt that as such, keeping the pearl was more trouble than it was worth. However, from the fisherman's point of view at the time the pearl was found, considering the knowledge he had available to him as well as his ignorance of the future, there is no reason he should have thrown it back. Doing so would have jeopardized his family's possible financial security.

The two examples, that of the quiz and of the fisherman, involve the same paradoxes. These arise as a result of implications from various points of view.

KelseyMilliron said...

I am am truly baffled by this scenario and am really having a hard time grasping this. I feel like on the last day if the quiz were to be given then it would not be a surprise because as the sharp student response said that there would be no more days left so we would know it was the last day. But I feel like on the second to last day it could still be a surprise because you don't know if it is going to be given the last day or on that day. It would not be as huge as a surprise as you would have a 50/50 chance on guessing the correct day but you still would not know. Also if the quiz is on a topic you do not know about then technically I would consider it a surprise quiz on any day.
I guess that last remark I made contradicted with the teachers saying the students will have no way of guessing on what day the exam will be.
Obviously teachers can give surprise exams and I think that any day the exam would be given (other than the last day) would be a surprise.