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Tuesday, March 14, 2006

The Surprise Quiz Paradox

Kaimi Wenger raises some issues about the subjectivity of grading here and here.  Let me offer an entirely different puzzle about student examinations that has long stumped philosophers and may interest or amuse Prawfsblawg readers.  Here's my formulation of what is often called the "Surprise Quiz Paradox":

On Friday, a high school teacher announces to his class that there will be a surprise quiz some time during the next week.  "By surprise," he states, "I mean that, the night before the quiz, you will not be able to deduce that the quiz will be the next day."  A clever student raises his hand and says, "Well, the exam definitely cannot be on Friday because on Thursday night, we would be 100% certain that the exam would be the next day.  But then, the exam cannot be on Thursday either because, knowing that it cannot be on Friday, on Wednesday night we could deduce that the exam must be on Thursday.  By similar reasoning, it is impossible to give the exam on Wednesday, Tuesday, or Monday. Therefore, your claim that you will give a surprise quiz next week is clearly false."

The next week, the teacher administers a quiz on Tuesday, much to the surprise of all the students.  Do you agree with the student's claim that a surprize quiz absolutely cannot be given on Friday?  If so, where did the student's reasoning go astray?  Feel free to post your answers in the comments section.  Be warned, though.  This problem is harder than it looks.  (For more puzzles and some trivia, check out posts by Kevan Choset at the Volokh Conspiracy.)

Posted by Adam Kolber on March 14, 2006 at 10:43 AM | Permalink


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The paradox of surprise test is based on the backward counting starts from Friday as it is given in the student's elimination argument.The definition of surprise test is that the exact date of the test should note be known to the student beforehand.Now,most of the recent discussions on this paradox is revolving round the day of prediction.One cannot argue against the student's argument by showing that on Wednesday night or some other day preceding Wednesday, one cannot predict the day of the test. These sorts of solutions are not hitting at the problem, given the definition of surprise test and the elimination argument.The paradox is still there because given the scenario,the student can predict the day at least on Thursday night.The solutions that i mentioned above are just beating around the bush and not looking at the problem with precision

Posted by: Robin Luke Varghese | Nov 19, 2008 10:38:52 AM

@MR: "Seems like the problem is that the logic doesn't hold for the whole wee. Yes, it's true that once Thursday passes, the students will know that the quiz is on Friday. But that's where it stops.

Once Wednesday passes, the quiz could be either Thursday or Friday. Thus, the students won't know which night to study for. Let's say they follow the logic that it cannot be on Friday because they will know. So they all study Thursday night. Surprise, surprise, there is no quiz on Thursday - the time is wasted. Of course, they now know the quiz must be on Friday."

This is false. Because you say that on wednesday night they could think that it can either be friday or thursday. But since the teacher specifically stated that they cannot deduce the night before, if the quiz with be the following day, it could never be friday. So on wednesday night, they cannot be thinking that it can either be friday or thursday, since friday is completely out of the picture.

Posted by: jeeper | Aug 10, 2008 5:06:58 PM

after reading everones post i have come to my uneducated dicision based on what was Posted by: Posted by: jk | Jan 16, 2008 5:37:50 PM

makes the most sense to me,

but if you had to narrow it down to trying to figure out what day the teacher has planned the test to be i would go with the Post by: cvr | Feb 28, 2008 7:05:18 PM

Posted by: zoeskyfarm | May 22, 2008 11:21:33 PM

provides the answer using a spatial analogy

the only way the statements of the teacher can be consistent is if they are understood as meaning that the test will come as a surprise on all days except if it happens to be given on friday.

formulated this way, there is no paradox, and a test on any day other than friday will be a surprise test. hence our intuitive sense that a surprise test is possible.

Posted by: cvr | Feb 28, 2008 7:05:18 PM

It seems to me the only way the teacher could possibly have surprised the students would have been to keep his trap shut about the quiz. How much sense does it make to say, "I will be surprising you with a quiz?" It's like saying, "Sometime this evening you will be surprised with a birthday party." Expectation is created, no matter what. And this has nothing to do with logic or maths, unless one embarks on a discussion of where logic and maths intersect with processes of thinking and perhaps the neurological underpinnings of communication and expectation. So, for fear that I am drifting beside the point, it's obvious that because the teacher has prepared the students for a test - which he for some reason has dubbed a "surprise" - they ipso facto cannot possibly be surprised by it, in which case what we have is not a logical paradox but a mildly interesting exercise in armchair psychology.

Posted by: jk | Jan 16, 2008 5:37:50 PM

Let me rephrase the problem as follows:
Axiom 1. On exactly one day out of the next n days, there will be a quiz.
Axiom 2. For each day k, given the results of days 1,2,..., k, there does not exist a proof that the quiz will be on day k+1. (This is the "surprise" requirement.)

Claim: The quiz cannot occur on any of the days.

Proof: If the quiz were on Friday, then given the results of Monday through Thursday (no quiz on each), a proof would exist on Thursday that the quiz is on Friday, as there must be a quiz (Axiom 1). Since Axiom 2 states that no such proof will exist, we conclude the quiz is not on Friday. Note that this proof is valid without actually knowing the results of any day. Thus, this is "known" at time 0.

Inductively, suppose we have proved (at time 0) that the quiz cannot be on any of the days k+1, ..., n:

Subclaim: the quiz cannot occur on day k.
Proof: Suppose the quiz occurred on day k. Then after knowing the results (no quiz) for days 1,2,...k-1, we know that the quiz cannot occur on any day other than k. By Axiom 1, it must occur on day k. But then we have a proof using only the results of days 1,..., k-1 (remember, the other days are eliminated by the inductive step at time 0, so we need no results to eliminate them) that the quiz occurs on day k, contradicting Axiom 2. Thus, the quiz cannot occur on day k.

Thus, we conclude that the quiz cannot occur on any day, contradicting Axiom 1.

The punchline: You can call it a "paradox" if you want, I call it an inconsistent axiomatic system. Who said things in math weren't falsifiable :-)

Posted by: Ben Leitner | Oct 5, 2007 5:47:07 PM

I thought about it a lot and i think that the surprise quiz can't be held...

for more details, refere to my solid reasoning at http://agarwal-rohit.blogspot.com/2007/02/surprise-that-never-was.html

Posted by: Rohit | Jun 29, 2007 5:06:39 AM

FXKLM also made me wonder whether I am even dumber than I think I am so I googled the question. (BTW: This post has the highest ranking). The most persuasive "solution" seems to be that a paradox is created only if one thinks that logic is what is necessary to predict the future. However, predictions about future events do not depend upon logic but upon knowledge. Since our knowledge of all existing variables as well as any variables which will arise between now and the event is incomplete there will be a margin of error. However, if we disregard the facts and attempt to use only logic we will be living in a world that exists only in our thinking and we will indeed be surprised by a quiz on Tuesday.

Posted by: nk | Mar 15, 2006 7:57:45 AM

I remember too many disciplinarians to resist the opportunity to paraphrase their vaunted pedagogy. I like the mini-lecture on which I embarked when I recognized this is the moment to remind the students of the rising curve of probability of the test's being soon, though, as each testless day passes: certainty increases day by day until the test arrives, and that crescendo of imminence will drive increasing numbers of students to study. It reverses the regression a little, but is couched in more ordinary language; call it progression theory. I am glad for the absence of the drilldown Excel sheets with CAGR computations, though, now that I have left the market analysis field. Though it is a beguiling paradox you offer; nice when math and language are misced for the salubrious effect of confounding; it was nice to smile through.

Posted by: John Lopresti | Mar 14, 2006 10:11:18 PM

John: If you're trying to rephrase the problem so as to avoid the paradox, there's a much easier way to do it: There will be a quiz some day next week. Paradox resolved.

Posted by: FXKLM | Mar 14, 2006 10:02:02 PM

software requests post again. I wrote:

Taking a regression analysis approach:
Next week is a set, a linear progression of 5 days; the test will occur on 1 of the 1st 4 elements in the set M, Tu, W, Th, but I am not going to tell you which day; however, I have some other activities planned tentatively for early in the week, which, if those other activities take sufficient time, may necessitate my delaying the project which is required for me to devise the test's questions; but, surely by Thursday I will have the time to test the class, or perhaps at the very latest on Friday. The only day you will know ahead of time the test is the next day is if I am otherwise preoccupied and have had to delay the effort to make up the quiz; otherwise it will be a total surprise.
Now you must realize that as we near the latter part of next week the likelihood the test is coming increases; so, the smart student will study some Monday, more Tuesday, yet more Wednesday, and, ultimately, if Thursday arrives without the test, everyone may rest assured the test will arrive Friday and everyone who procrastinates until the very last day risks having to review all the materials during a cramming session Thursday night.
Further complicating your decisionmaking, is the materials are so voluminous that if you commence your pre-test review on Monday, you risk forgetting some of the minutia by Friday.
However, for students with thorough knowledge of the material, the test should be easy, and require little preparation. Yet, allow me to give you this caveat, that I have observed that many students in this section need to review more thoroughly before the tests, especially these imprecisely scheduled tests, so those of you who know you are in this group that needs to study several nights to have all the material fresh in your memory, please make the decision now to dedicate at least a few hours each night reviewing the materials so you do well on the test.
Furthermore, if I discern from questions in class that the same group of students who tend to be less current in their background studies continue to reveal by their participation and question framing that they persist in this high-risk approach to tests when the tests are not scheduled precisely, then I will spring the test at the optimal time to catch these lagging students when they are least prepared.

Posted by: John Lopresti | Mar 14, 2006 9:38:18 PM

Ok, I take back my variation. (Although our illusion of immortality is a wonderful psychological and philosophical subject for another thread).

I stick to my "pedagogical". "There will be a quiz by and including next Friday but I am not telling you which day because I want you to study every night. If the subject is hard, I will likely give the quiz at the end of the week so you will have studied more. If the subject is not that hard and I am a good teacher I will give you the quiz early in the week so you will go on with your lives." Any "surprise" here is a result of the students failing to judge the teacher's assessment of the difficulty of the subject. Which, I still maintain, is not a surprise.

Posted by: nk | Mar 14, 2006 6:09:21 PM

nk: That's true only if the problem states that students won't know at the beginning of the week when the test will be held. But it doesn't. The problem states that on the night before the quiz is given, no one will know that the quiz will be given the following day.

The variation is different because there is no defined end point in our lives as there is in a week. We can reason backward from the last day of the week. We can't reason backward from the last day in our life because we don't know beforehand when that will be.

Posted by: FXKLM | Mar 14, 2006 5:41:07 PM

Sorry. It should read "The students have been warned that there will be a quiz next week ..." Curses. (Smile)

Posted by: nk | Mar 14, 2006 4:40:34 PM

Where's the surprise? The students have been warned that there will be a quiz next but the teacher refuses to tell them on which day because he wants them studying every night, beginning on Sunday and going through Thursday (although by Thursday they should all know the subject thoroughly). There is nothing paradoxical, philosophical or statistical here. Pedagogical, yes. Although there is the danger of de-sensitization. If the teacher does it too often the students may just choose to go on with their lives and take their chances on when the quiz will be.

A variation: We will all die. Most likely, we will not have warning even the night before. OK?

A lot of paradoxes and a lot of philosophy for that matter are simply the result of our curse of language. I have come to wonder whether the myth of the Tower of Babel had to do with a multitude of languages or simply language at all. I.e., that the people started dealing with words and not with bricks and mortar.

Posted by: nk | Mar 14, 2006 4:37:57 PM

A surprise quiz could be given on Friday. The quiz on Tuesday was not the surprise quiz that the professor warned about, but the students believed it was. They could then have been surprised by the actual surprise quiz if it had been given on Friday.

Posted by: Ilene | Mar 14, 2006 3:28:19 PM

I agree with the comments. What throws people off here is the idea that probabilities can't change over time. But the Monty Hall problem demonstrates how that's not true.

Posted by: Bruce | Mar 14, 2006 3:07:36 PM

FXKLM: I'm sure that both tellings must be off, if only because there is so much literature on the subject. I.e., many philosophers are convinced that there is some there here, so maybe we're not looking at the problem in the light most favorable to them.

Posted by: billb | Mar 14, 2006 3:03:07 PM

billb: Although the problem is generally regarded as a paradox, I think the telling of this particular version of the problem is a little off.

In this version, the paradox is that the students are surprised by the quiz even though the student's reasoning suggests that there could be no surprize quiz. That's not quite right. The student reasons that since the test can't be given on a particular day without the students knowing about it the day before, there must not be a quiz because if there is a quiz on any particular day, the terms of the surprise quiz are violated. The student fails to realize that a complete lack of a quiz also violates the terms of the surprise quiz. If the student's reasoning had been correct, he would have recognized the paradox then. It's not really a paradox when the students are surprised by the quiz on Tuesday. They came to the conclusion that there would be no quiz through faulty logic.

Posted by: FXKLM | Mar 14, 2006 2:58:05 PM

The philosophers that make up these sort of paradoxes must have answers to the sort of solutions that FXKLM, MR, and pk raise above, though the link that you provide doesn't really shed any light on them. It seems to me that "surprise" is sufficiently vague to allow those who are emotionally attached to this paradox to continue to call it a paradox after laymen have already satisfied themselves that there is no paradox to worry about.

If the teacher simply draws the day for the exam randomly from a hat, then 4/5 times the exam is offered, the students will be "surprised" by it, and 1/5 times the students won't be. However, they don't have any way to know for certain not to study until Thursday night since 80% of the time the exam will come sooner than Friday. Therefore the students don't have any choice other than to study each night for an exam the next day, even though, in the end, they may not actually be "surprised" by actual date of the exam. This is enough, in my layman's (i.e. non-philosophers's (query whether having a PhD makes me a defacto philosopher)) mind to settle the problem. That is, the problem isn't whether the teacher can give a surprise exam, but whether the students can delay studying until the last possible night, and I think I've demonstrated that they cannot.

Posted by: billb | Mar 14, 2006 2:24:50 PM

I think this problem gets overtheorized. The student is correct about Friday. On Thursday night, there would only be one day left. The student erroneously extrapolates backward, *taking as a given from earlier in the week* information only obtainable on Thursday night. You can't bootstrap yourself back that way.

Also, that definition of surprise quiz is silly. Any quiz with the word "BOO!" on the front ought to qualify.

Posted by: Eh Nonymous | Mar 14, 2006 2:12:22 PM

The student is correct in that a Friday exam is not possible, but it is still possible to have a surprise exam without introducing the possibility of no exam. The key is that the student starts with the knowledge he has as of Thursday night and then reasons backwards. In other words, by Thursday night he knows that an exam has not been given on Monday, Tuesday, Wednesday or Thursday. But, on Sunday night, he does not know whether an exam will be given on those days.

Posted by: pk | Mar 14, 2006 11:48:28 AM

Seems like the problem is that the logic doesn't hold for the whole wee. Yes, it's true that once Thursday passes, the students will know that the quiz is on Friday. But that's where it stops.

Once Wednesday passes, the quiz could be either Thursday or Friday. Thus, the students won't know which night to study for. Let's say they follow the logic that it cannot be on Friday because they will know. So they all study Thursday night. Surprise, surprise, there is no quiz on Thursday - the time is wasted. Of course, they now know the quiz must be on Friday.

But let's now go back to Monday. Monday passes, no quiz. Do they study? Well, the quiz could be Tuesday, Wednesday, Thursday, or even Friday. Even if we rule Friday out, we can't do that until we get to Thursday, so it is still a possibility on Monday night.

Seems like a Baysian probability issue to me. The probability that the test will be on Friday given that we are on Thursday is very different than the probability that the test will be on Friday given that we are on Monday.

Posted by: MR | Mar 14, 2006 11:42:33 AM

The students were surprised by the test on Tuesday because they had come to the conclusion that there would be no test. If we allow for the possibility that there might not be a test (as the students apparently did), it becomes possible for the test to be a surprise on Friday, which makes it possible for the test to be a surprise on any other day. Once the student recognized the possibility of no quiz, the paradox was destroyed.

Posted by: FXKLM | Mar 14, 2006 10:57:11 AM

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