Tags Math
This paradox, IMHO, is far simpler than the Two Envelopes, though at first is also very confusing.

I'll approach the solution from two sides.

First, the mathematic, probabilistic side. The days of the week are 1-5 for Monday-Friday. Furthermore, lets denote se(N) as the event that there's a pop quiz on day N, and no_se(N) as the complementary event. Given that the quiz must happen during the week, it is trivial that: P(se(1)) + P(se(2)) + P(se(3)) + P(se(4)) + P(se(5)) = 1 Therefore: P(se(5) | no_se(1), no_se(2), no_se(3), no_se(4)) = 1 In words: there must be a pop-quiz on Friday, given that it wasn't on any earlier days. The students concluded from this that there can't be a surprise-quiz on friday. However, A-PRIORI and A-POSTERIORI probabilities are mixed here. Students make a claim (there can't be pop-quizes on Friday) A-PRIORI (before even Monday happened), while this is true only given that Monday-Thursday happened (A-POSTERIORI). No one said: P(se(5)) = 0 We only said: P(se(5) | no_se(1), no_se(2), no_se(3), no_se(4)) = 1

From the practical (real life) side: the whole term "pop quiz" is badly defined. Where is the surprise ? In the fact that the students don't know ahead on which day the exam will be. So, assume one smart-ass student is sitting home on Saturday eve, and contemplating. Can there be a pop quiz tomorrow ? According to his reasoning, there can not. But say the teacher decided beforehead that there is an exam on Sunday. The student won't study, and the exam will "surprise" him in this sense. By saying that there is a pop quiz next week, the teacher achieves his goal: the students should study every night, as they can't really predict whether there will be an exam on the next day, or not. Sure, on Thursday evening the students will "predict" that there will be an exam tomorrow, but so what ? They didn't know it on Wednesday. They knew that given that tomorrow (Thurs) there's no quiz, it will surely be on Friday. But who told them there won't be a quiz tomorrow ? So they studied anyway.

To conclude - yes, there CAN be a pop-quiz on Friday.


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