Here’s an irritating little conundrum.

Imagine you are a student in a philosophy-of-logic class that meets five days a week. On Friday the teacher announces that there will be a surprise quiz one day next week. As you are leaving class it dawns on you that it can’t be next Friday, because if you get all the way to Thursday without having been given the quiz, you will know that it has to be coming on Friday, and therefore won’t be a surprise. So Friday’s definitely out. It has to be Monday, Tuesday, Wednesday, or Thursday. But, having realized that (and of course, your professor, being an expert logician, will have realized this as well), then if you get to the end of Wednesday without having had the quiz, you’re going to know that it has to be Thursday, because we’ve already conclusively established that you can’t give a surprise quiz on Friday. But this means you can’t give a surprise quiz on Thursday, either! So having ruled out Thursday and Friday, it’s got to be Monday, Tuesday, or Wednesday. But if Wednesday is the last possible day, then this really means it can’t be much of a surprise on Wednesday, either, by the same reasoning. So Monday or Tuesday, then. Actually, it’ll have to be Monday, because Tuesday is now the last available day, and if you haven’t had it Monday, you can expect it for sure on Tuesday. But if it has to be Monday, then it’s no surprise at all.

You are forced to conclude that your teacher simply cannot give a surprise quiz next week; he probably was just hoping that some of the students would see this elegant proof. So you don’t even bother to study, and are thoroughly shocked on Thursday when the quiz is presented.

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  1. Kevin Kim says

    I’m terrible at thinking in mathematical terms about probability, especially at setting such problems up in a mathematically coherent way, so please feel free to poke holes in what I’m about to write.

    It seems to me that the probabilities vary depending on one’s temporal perspective. Let’s say it’s the Sunday before the week in question. From Sunday’s vantage, the chance that there will be a quiz on Monday is 20%, i.e., one in five. If the quiz doesn’t occur on Monday, then the chance of a quiz happening on the remaining four days is 25%, i.e., one in four. If the quiz doesn’t happen on Tuesday, the probability of the quiz’s taking place on one of the remaining three days goes up to 33.333…%, i.e., one in three. If there’s no quiz on Wednesday, then from Wednesday’s vantage, there’s a 50% chance of a quiz either Thursday or Friday. It’s only from Thursday’s vantage — assuming (1) no quiz occurs on Thursday and (2) the quiz hasn’t already happened on an earlier day — that the chance of a quiz on Friday is 100% (if the quiz has already happened, then the probability of a quiz on Friday would be 0%).

    I would say that the student who is surprised by the Thursday quiz has set the problem up incorrectly by working backwards through time instead of forwards. Not being a math whiz and knowing nothing about the laws of probability (that phrase being suspiciously oxymoronic), I couldn’t begin to explain why temporal sequence might matter in the calculation.

    As a matter of lateral thinking, I would also note that “surprise” might not denote something temporal at all: it might simply mean that the nature of the quiz will be unanticipated. The teacher might say, “You’re having a surprise quiz next Thursday: you won’t know what’s on the quiz, and it might not be in a format you easily recognize at first. That’s the surprise.”

    It occurs to me, too, that in this case, probability is a matter of knowledge/perspective. The teacher might know full well that he’ll be giving his quiz next Thursday. In that case, from the teacher’s perspective, the chances of there being a Thursday quiz are 100%. Strangely enough, this also means that, from the teacher’s perspective, the quiz isn’t a surprise quiz in the sense of “temporally unanticipated.” For the teacher, there is no surprise. Perhaps he is being disingenuous in calling it a surprise quiz…?

    To ensure absolute surprise, the teacher could write up his quiz, then tell the students: “I might be giving you a surprise quiz next week.” Then, at the beginning of each class, the teacher might flip a coin or roll a die or ask a random student to pick a number between 1 and 10, inclusive. In that case, neither the teacher nor the students would know, on any day, whether there would be a quiz; they would know only that the chances of such a quiz occurring would be non-zero. They would also know that they might be able to escape the week without taking a quiz at all, which is why the teacher would have said, the previous week, “I might be giving you…”


    Posted January 11, 2008 at 11:39 pm | Permalink
  2. Malcolm says

    Hi Kevin,

    A splendid response, and far more than I expected to see from anyone.

    The probabilities do vary as you say, and you are not alone in raising this objection. It is still the case, though, that a quiz with a probability of 100% is no surprise, so Friday is indeed ruled out. You simply cannot give a surprise quiz on Friday. The only surprise in that scenario might be getting to the end of Thursday without having been given the quiz yet.

    Others have also pointed out the forward-backward issue you raise. There is an inherently forward-facing temporality in the very concept of a “surprise”, however, which is that an event at time t+1 was not foreseen by observers at time t, so I don’t think one can brush aside the problem in this way.

    The curious part of this example is the way it the particular cognitive state of the “surprisee” – namely whether he has or has not worked through the reasoning – cancels out. To a student who hasn’t made the regressive calculation, a Wednesday quiz will be a surprise, for the conventional reason that he simply doesn’t know what day it will be. To one who has worked through the logic, a Wednesday quiz will still be a surprise, because he has convinced himself a surprise is impossible.

    Googling “Surprise Quiz Paradox” turns up some interesting discussions.

    Posted January 12, 2008 at 6:44 pm | Permalink
  3. Kevin Kim says

    Thanks. Could you explain what you meant in your paragraph beginning, “Others have also pointed out…”? I’m not sure I understood the point. I’d agree that there’s an inherently forward-facing temporality in the concept of “surprise,” but what I don’t get is the final concluding claim that “I don’t think one can brush aside the problem in this way.” My own reckoning was from the forward-facing perspective (starting from Sunday and looking forward in time), as opposed to reckoning backward from Friday, so if I brushed anything aside, it was the backward-tending approach. (I think.)

    Also, re: “You simply cannot give a surprise quiz on Friday,” I’d agree this is true, but only from the Thursday vantage, and only after class is finished that day. If by “surprise” we mean “an event whose occurrence at some future time T is partly or totally unanticipated,” then it’s true that, on Wednesday, we still don’t know (i.e., can’t anticipate) whether the quiz will occur on Friday; all we know is that there’s a 50% chance of the quiz occurring on either day. On Wednesday, therefore, we can legitimately say to a classmate, “I don’t know… we might be in for an ugly surprise on Friday.”

    NB: I say “partly or totally” because I think there are degrees of surprise, such as when one suspects, but isn’t sure, that people will throw one a surprise birthday party, as opposed to those times when something completely unexpected occurs.

    Actually, the more I think about it, the more I disagree with the notion that, if we know we can’t be surprised on Friday, we can’t be surprised on Thursday, and so on backwards through time. Thursday strikes me as a special threshold. If we’re reasoning backwards and we hit Wednesday, then once again, from Wednesday’s vantage, both Thursday and Friday are possible quiz days.

    It seems that the work of better minds than mine would be to demonstrate how and why the “backwards temporal reasoning” (retrotemporal ratiocination?) process is illegitimate.

    Aside: in looking through some of the comments (after doing the Google search you recommended) I saw one that interested me: the commenter contended that such paradoxes arise from the use of logic instead of knowledge. Perhaps for the very reason you mention in your comment above (“the reasoning cancels out”), logic is useless in such cases, and as the teacher is the only one who knows what day the quiz will be, probability is not even an issue. In fact, objectively speaking, even though the student lacks knowledge of when the quiz will take place, there is still — even from the student’s unknowing perspective — a 100% chance that the quiz will take place on the day appointed by the teacher.

    I have no idea where I’m going with any of this, so I’ll stop here. Heh.


    Posted January 13, 2008 at 12:32 am | Permalink
  4. Malcolm says

    Hi Kevin,

    I can see how my remarks were confusing (pretty much everything about this paradox is confusing). What I was getting at about not brushing away the problem is that student who works it all out, concludes there won’t be any quiz, and is then surprised, hasn’t done anything backwards; he has iterated over the days of the week forwards, but seeing upon reaching the end of the set that that can’t be the day of the quiz, has to trim day after day from the set of possible candidates. What I meant was that you can’t brush away the problem (I think!) just by saying that it arises from working backwards.

    This is also known as the “Unexpected Hanging”, and a counterargument to the “you may know on Thursday, but you won’t know on Wednesday” objection you raise is given in this Wikipedia article:

    The argument that first excludes Friday, and then excludes the last remaining day of the week is an inductive one. The prisoner assumes that by Thursday he will know the hanging is due on Friday, but he does not know that before Thursday. By trying to carry an inductive argument backward in time based on a fact known only by Thursday the prisoner may be making an error. The conditional statement “If I reach Thursday afternoon alive then Thursday will be the latest possible date for the hanging” looks about as appealing as last week’s lottery ticket at half price. The prisoner’s argument in any case carries the seeds of its own destruction (execution?) as if he is right then he is wrong, and can be hanged any day including Friday.

    The counter-argument to this is that in order to claim that a statement will not be a surprise, it is not necessary to predict the truth or falsity of the statement at the time the claim is made, but only to show that such a prediction will become possible in the interim period. It is indeed true that the prisoner does not know on Monday that he will be hanged on Friday, nor that he will still be alive on Thursday. However, he does know on Monday, that if the hangman as it turns out knocks on his door on Friday, he will have already have expected that (and been alive to do so) since Thursday night – and thus, if the hanging occurs on Friday then it will certainly have ceased to be a surprise at some point in the interim period between Monday and Friday. The fact that it has not yet ceased to be a surprise at the moment the claim is made is not relevant. This works for the inductive case too. When the prisoner wakes up on any given day, on which the last possible hanging day is tomorrow, the prisoner will indeed not know for certain that he will survive to see tomorrow. However, he does know that if he does survive today, he will then know for certain that he must be hanged tomorrow, and thus by the time he is actually hanged tomorrow it will have ceased to be a surprise. This removes the leak from the argument.

    Frankly, this one still has the better of me; I have yet to encounter a satisfactory account of just what the problem is here. I recall confronting this many years ago, long before the Internet resources we have now were available, and not getting to the bottom of it then — and it just popped into my head for some reason last night, and I thought I’d post it while I was trying once again to get my arms around it. But today I’ve had no time for dwelling on it, or following up on what the philosophers have done with it (and the fact that it has had such a fuss made over it shows that it is far from trivial, I think).

    Posted January 13, 2008 at 1:33 am | Permalink
  5. patrick says

    If he tells them there will be a quiz at any time next week, it is thus no surprise anyway…Figuring out the day is not meaningful as the surprise is already ruined.

    Posted January 13, 2008 at 2:40 pm | Permalink