A professor tells his class that he will give them a surprise examination sometime next week.
For our purposes, an examination is considered a "surprise" if on that day, the class is not 100% sure that they will get an examination. (If they only suspect it, the examination can still be a surprise.)
Having given the class this warning, is it still possible for the professor to give them a surprise examination next week? (Assume that he will not give them an exam if it will not be a surprise.)
Suppose it's Friday and the "surprise" test hasn't come. The professor walks in, and starts a lecture and lectures until the end of the class. With 2 minutes to go, someone says, "Hey, what about the pop quiz?" The professor mumbles something out about not having the questions ready. Finally he says, "We'll have a surprise test next week!" Just kidding! :)
If a professor says, "We will have a surprise test next week." I believe what he really means is, we will have a test on any day Monday - Friday assuming that's the work week. The initial probablility of any day being 20%, and steadily increasing to 100% by Friday. When you exclaim a surprise test within a given time frame, the surprise is somewhat diminished.
For instance, with past history of human life spans, I know I'll be dead sometime between now and the next 90 years or so. If I had no concept of mortality or human life spans or even time for that matter, death may be a grave surprise to me upon its arrival. I might not ever be prepared for such a pop quiz like that. Knowing these things, if I died tomorrow, it might be somewhat of a surprise. However, if I die at 93 years old, it isn't too much of a surprise or it shouldn't be.
So, I believe the professor proclaiming the pop quiz was equivalent to telling the class that they were all going to die within the next 100 years. That's not really a "surprise", but it's the way I see it.
Next week.
