“he will not know when the test is coming.”
This phrasing is not that specific. One way to make it specific is if the teacher instead offers
“Every morning I will let you wager on whether the test will occur with the following odds: you may bet 9 dollars to win ten dollars if the test happens that day, nothing otherwise. you may bet up to $9000 each day. The test will be a surprise, so you won’t be able to make money.”
The student then picks a bet size each morning, after which the teacher decides whether or not to give the test (except that on the last day, they must give the test if they haven’t yet. ) This is a zero sum game of perfect information, and one which the student wins by betting 9000 on friday, 900 on thursday, 90 on wednesday etc which in this formalism is the operationalization of not being surprised. The teacher is wrong, although they only are wrong by a couple of cents.
However, if the formalization of the problem is changed to where the student must make a binary choice each day (bet 9 dollars or nothing) then they are unable to profit and are therefore surprised. The teacher is right.
Make another tiny tweak: each morning the student writes their bet on a piece of paper, which must be either 0 or 9, then the teacher reveals whether the test is that day, then the student turns over their paper and money changes hands. With this version, the student wins again (proof exercise for the reader: the students winning strategy is to randomize whether to bet and 10x their odds of placing a bet each day. why is this winning?) This version is imho the nicest: In the first version, if the student plays optimally the teachers choice doesn’t matter, in the second version the teacher’s optimal strategy is simply to always give the test on the forst day the student doesn’t bet or on friday; but in the final version the teacher’s optimal strategy is very nearly to give the test 90% of the time every day, leaving the student indifferent as to whether to bet at 90% odds (i.e. the student has a correct 90% subjective provability that the test will be today, every day)
Given that the answer differs between reasonable formulations, the original problem is in my opinion underspecified.
Suppose on each day the student rules out all subsequent days, concluding that the test must be today since that's the only day left, but stops reasoning there and goes to class having "proven" that there will be a test that day.
Yeah, that's fair. I like my solution to the paradox over the logical school one since it focuses on the absurdity of a causal link between what the student expects and what occurs in reality as opposed to merely "surprising" being under-defined, but they're both quite interesting.
This is a crosspost from my blog post.
In this post, I’m going to resolve the surprise test paradox.
The surprise test paradox is as follows:
A teacher tells a student that he’s going to have a test this week, but that he will not know when the test is coming.
Upon hearing this, the student realizes that he will not have a test on Friday, since, if the test were on Friday, he would know that morning that the test was going to occur.
But, since he knows that the test won’t occur on Friday, he realizes that he also won’t have a test on Thursday because, on Thursday morning, he would expect for the test to occur since he would already know that it’s not going to occur on Friday.
Then, using similar logic, he also deduces that the test also won’t occur on Wednesday, Tuesday, or Monday, and that, as such, he shouldn’t expect for the test to occur at all.
Glad that he isn’t going to have a test, he walks into class on Wednesday and is, of course, handed a test.
I used to think this was a paradox since it seems like the student’s logic is correct and, yet, it leads to a conclusion that guarantees that he will not know when the test is coming by causing him to expect for no test to come at all.
I now no longer think that it is a paradox.
When the student concludes that the test is not going to occur on Friday, he, in fact, makes it possible for the test to occur that day since he, now, no longer expects it. As such, the student made a reasoning error by failing to take into account the fact that his expectations determine whether or not a test occurs. If the student were instead reasoning properly, he should have realized that, each morning, he should expect a test to occur that day, since, if he expects it to occur, it will not.
So, in reality, the surprise test paradox is not a paradox at all. If each morning the student expects to have a test, he will never receive one and the teacher’s statement will be false. If the student doesn’t expect to have a test on a given morning and then receives a test later that day, the teacher’s statement is true.
What makes the situation strange is that, each morning, the student should expect to have a test despite the fact that, if he expects to get a test, he “should” also expect not to get a test since expecting to get a test guarantees that he will not get a test. Although this is a strange state of affairs, it is not paradoxical because the first “should” and the second “should” are two different kinds of “shoulds.” The first “should” is a should based on what he ought to do to avoid being executed. The second “should” is a should based on what he ought to do to be logically consistent. This is only a paradox if one believes that individuals ought to be logically consistent in all situations, which this “paradox” clearly reveals is not the case.
I’m not sure whether the paradox has been resolved in this way by others in the past, but I thought I’d share it with you guys since it’s quite an interesting philosophical conundrum.