# Story: A surprise quiz

When the regents of the university hear that it was possible to get a doctorate of philosophy without even taking a single class in philosophy, they wrote a new rule: "All Ph.D. candidate must take and pass at least one course on philosophy." This gladden the hearts of the philosophy department, but annoyed the mathematicians since they dreaded the prospects of having to write actual sentences!

It came to pass that Joe Math, a student of probability, was starting to take the required philosophy. The professor started out with the usual admistrivia about the course:

"This is the remedial philosophy course for natural scientist. Thus, we will allow you to stay in the class as long as necessary to learn the material. At the end of each semester, if you haven't passed, you will simply roll your points over to the next semester and no grade will appear on your transcript. This avoids having the stigma of getting an F on your transcript."

"The grading will be based on written themes (one per week), exams (3 per semester) and pop-quizzes. Grades will be the usual, 90% for A, 80% for B, etc. As you are all graduate students, you have to get at least a C to pass."

"To show you the power and clarity of logical thinking, I will give an A to any student that catches me in a falsehood. In all the years of teaching, no student has ever managed to pass via this route."

One of the students raise her hand and asks, "How often will the pop-quizzes be?"

The professor answers, "Each week there will be one pop-quiz. (For you non-native English speakers, pop-quiz means a quiz that you will not know about ahead of time.) If you feel that you have figured out my pattern for giving pop-quizzes you are welcome to bet with me before class about whether I'll give one that day. Since you are supposed to be sure, I will require you to give me 9 to 1 odds. (In other words, you should only bet if you are more than 90% sure of winning.) Since there are 5 days of class a week, you will have to do significantly better than the base rate of 20% in order to improve your standing."

At the mention of betting the probabilist starts paying attention. "How much are we allowed to bet?"

"Oh, I didn't mean money - - I meant points for the class. You can bet up to the number of points that you have earned so far in class. In fact, I value induction so well, that if you manage to leverage a close to perfect grade in the class to one where by you have more than 105% total, I'll give you an A+."

A few days later the first pop-quiz was held. The probabilist who obviously hadn't done the reading (and in fact hadn't even gotten around to buying the book yet) did lousy on it. But the professor felt charitable and gave him 3/10 points. He attached a note saying, "You will have to study if you want to pass this course."

On reading this, the probabilist became very happy.