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.)