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The Unexpected Hanging

The Paradox:

A teacher tells the class that they are to have an examination next week. The teacher also tells them that they will not know which day it is to be held on, for it is to be a surprise.

One of the students surmises; “There can’t be such an examination” and outlines as follows:

Assuming an exam will take place next week, if we have not had an exam by the end of the day on Thursday we can deduce that the exam will take place on Friday. We would know to expect the exam on Friday and therefore it wouldn’t be unexpected. Likewise, if on Wednesday afternoon we had not had an exam (and knowing that an unexpected exam can’t be held on Friday), we can deduce that an unexpected exam can’t be held on Thursday. The same argument holds for Wednesday, Tuesday and Monday and therefore you can’t hold an unexpected exam.

The next week, to everyone’s surprise, the teacher throws an exam on Wednesday.

The Solution:

The student’s argument entails expecting an exam every day. The paradox occurs because what is required to “expect” an exam is never defined. If one can expect an event to occur because it is possible (as the student does) or whether one can expect an event to occur because it must occur (as the teacher does) needs to be defined.

At the start of the week there’s a 1 in 5 chance that the exam will fall on any given day (it’s possible on any day). If there’s no exam on Monday then there’s a 1 in 4 chance that the exam will fall on any of the remaining days. This continues likewise until Friday (where the exam must take place if it hasn’t already).

The students argument is based on “if it’s possible there’s an exam today then we’ll expect it”, whereas the teacher’s argument is based on “they can only expect the exam if they know one must take place”.

So how could the teacher throw the exam on Wednesday and surprise the students? The students’ didn’t realise that their argument entailed them to expect the exam on every day. They have much to learn!