The surprise exam paradox

[billiams]

The paradox is a lot easier to see if you narrow it down to just one day.

The professor says, I will give you an exam tomorrow, and it will be on an unexpected day. The students realize that this is impossible, so if it is to be unexpected, it can't be tomorrow. Then they are happy and think the evil professor made a promise he couldn't hold, and don't prepare for the test. However, the next day, they have the exam. And all are surprised.

I agree that's an element of the paradox. But certain expressions like commands for example aren't truth functional and our linguistic conventions regarding such expressions reflect this. If the teacher said, "I will give you the test, and you will love it." The former he can do; the latter he can command, or hope for, but it's not a truth functional claim.

Similarly, surprise can not be commanded, or if it is , it is a wish or hope, but not fact that can be reliably averred. If I told you there is a cupcake in the fridge, and when you see find it, you will be surprised. You ask, "Why, is a special cupcake? I say, "if you like Hostess, it is, but otherwise no." "You will be surprised that there is a cupcake there at all." You would not be surprised.

Now, maybe you know me to be a glutton, or a liar, or both, but assuming I'm truthful and not a glutton, and you believe these things about me, you just won't be surprised. I haven't lied, because it was a hope, a naive, stupid hope that you would be. It's not a hard fact the way the cupcake in the fridge is.

So if the teach says" You will take the test tomorrow, and you will be surprised" I would say you can make me take the test, but I'm sorry, if I know I'm taking the test, you can't surprise me by giving it. You can hope or command that I be surprised, I'm sure bad magicians hope they will amaze and astound, but they don't.

Of course, add an element of real unpredictability, and a use of surprise where it actual applies, and you get the paradox. But if the test occurs on Friday, you will know on Thursday, even if surprise is predicted, it just won't happen given the convention of surprise, Otherwise, you will really have no certainty ( there will be probabily -- see three card monty issue) when the test will be given, and no idea at all, on Monday.

--


--

This has already been mentioned in this thread, but isn't one relatively common interpretation of the surprise exam paradox is that it's a proof by contradiction that a certain set of premises is inconsistent?

That was my favoured interpretation for a while, but I realised it doesn't fit... if the premises were inconsistent, it would be impossible for them to all be true at once (by definition), and yet at the end of the story, there was an exam, during the specified time period, and the class was surprised.

The claim that the premises are inconsistent are at the heart of the paradox. Specifically: 1) The exam will be given on one of the five days of a week running consecutively from Mon to Fri ; 2) The students will be necessarily be surprised on the day of the exam; 3) Enthymatic premise, if one knows one will be tested on a particular day, one is not surprised. Say, the test is given of Friday, via disjunctive syllogism, the student knows the test will be given on Thursday, which based on 2 and 3, means he's not surprised, call this 4. 4 and 2 are inconsistent and you have a contraction S and not S.

I argue above that the definition and context of the term "surprise" creates this problem and will for any defined set where one knows only a limited number of events can take place within the subsets. That definition of surprise can not be met, provided the enthymeme 3, which is a basic convention of our language use.

--

If the set up is too confusing for George and the others, the paradox is just a case of misunderstanding, and there is no paradox, just one of those " What the hell was I thinking?" moments. People who don't grasp rules are bound to be surprised on a regular non-paradoxical basis.

Or in the alternative, the on the day of the exam part, which wasn't defined in the version I originally read, but was covertly assumed, is discharged, even if George and his class, follow the rules they will still be surprised ,even on the day of the exam, except Friday.

Or the teach is just violating 2, and again no paradox, just a teacher who screws up the rules and causes confusion/surprise in so doing.

[Yakk]

If the next poster in this thread has not actually read the thread, will I be surprised?

I wasn't surprised.

[billiams]

If the next poster in this thread has not actually read the thread, will I be surprised?

I wasn't surprised.

What was your expectation?

--

And to speak of the orginal set up-- the ruling out is deductive, not inductive -- disjunctive syllogism.

--

Quine did a relatively famous version, though he didn't claim to be the author.

I'll need to look through my notes to find out more on this.

Edit: The Oxford Journal, Mind, was host to Quine's paper on the subject, as well as a bunch of further discussion. If your college has a subscription to that journal, then the article title is "On A So-Called Paradox," 1953.

Quine basically concludes just what I did: That if the students are going to conclude "There is no test," then they must admit that this is a possibility from the start; and such a position voids their inductive rule, since, on Friday morning, they don't know whether there's either a test on Friday or no test at all.

Chapman and Butler, in 1965, offer an even more interesting take in the same journal, called "On Quine's 'So-Called Paradox'". If you look at my above post where I quoted silverhammerba, you'll see that I divided the kids' argument into two premises. Quine sorta denied (1); Chapman and Butler deny (2) by giving a good definition of "unexpected":

Let us first examine what the schoolmaster actually said. He made two statements: (a) that there would be an examination on one of the five afternoons of the following week; and (b) that the examination would be unexpected. We may define 'unexpected' in this context as meaning that by no process of valid logical argument can the boys at any time predict without contradiction the day of the examination....


...The argument eliminating the last day itself contains two arguments:
(i) The examination must be held on the last day because on the morning of the last day it is the only day left.
(ii) Because of (i) the examination is expected on the morning of the last day and therefore, by (b), cannot be held on that day.
Note that both arguments place the arguer hypothetically on the morning of the last day.

What has been proved by these two arguments is that, for a boy arguing on the morning of the last day, it is necessarily the case that the examination both is and is not held on that day. The conclusion that the examination must be held on the last day is just as warranted as the conclusion that it cannot then be held. Therefore the boys cannot predict, by a valid process of logical argument and without laying themselves open to contradiction, that the examination will be held on the last day. Therefore the examination, even if it is held on the last day, will be unexpected in the required sense.

I might just like this version even more than I like my/Quine's version.

There is still the question of when. The Quinian requirement has not truly been met. On Friday through valid argument, the students know this to be the day, The " cannot" is predicated upon an expressive assertion, that shows the flaw in making such an assertion the basis of factual or analytic claim. Imagine five identical jars and plucks a marble in only one. The Prof. makes the exact same assertions. It will be completely unexpected when the marble shows up (and moreover that will be the necessary and sufficient condition for a test to occur the following week. Each morning one jar is opened. On Thursday, the jar is empty. There would be no confusion, valid processes of logical argument would lead the students to conclude that the release of the marble ( hence the test) would occur the next day. The assertion that they would be surprised ( or find it unexpected in the defined way infra) would be defeated. See above for the cup cake example. My assertion that you will be surprised by the cup cake would count as rendering the finding the cup cake unexpected, given the high redefinition of the term by Chapman and Butler. of course, the actual use of our language would I aver actually be interpreted as I described: effectively a prescription, not a separate, coequal premise generating epistemic confusion, disabling our ability to logically determine that there is a cup cake. There is a category difference between the claims that Butler and Chapman don't appreciate, it seems to me.

Of course there is also the issue of how to unpack the portmanteau of the redefined " unexpected," which proves too much, namely that we would always be surprised ( again by adopting a counter-intuitive and implausible understanding of knowledge, ala Peter Unger in Ignorance). Even if all the Evil Prof said was " We will have a test tomorrow," and from that a student determined there would be a test tomorrow, this would be an attempt to deduce a putatively synthetic statement from a putatively analytic one, which can never be a deductively valid ( if that is to be the standard for " valid process of reasoning"") move. Enthymatic reasoning either covert or not and or stipulations ( which have no truth value) must be employed. Thus, there is no valid way to even determine with certainty (highly redefined) that the test will be on the day stated by the Professor, even the nicer evil one, who just tells you the day. Thus, we would always have the paradox if we adopted this reasoning, even in the ordinary cases where we are just told something with no bizarre conditions attached.

By proving too much, the argument Friday would remain unexpected really proves nothing at all. The paradox, so called, is no different from the ordinary epistemic condition we are in when the paradox is not play. So the apparent success of the paradox in rendering uncertainty is chimerical and is just a by-product of adopting a standards of reasoning we neither use nor accept in ordinary life or language use.

[billiams]

Clarifying thought experiment using the hanged-man formulation: Judge sentences man to death for sins against logic. The Judge states that the date will be a complete surprise ( or that it will be unexpected), but he's willing to put his money where his mouth is, so if the man can predict the scheduled date of the hanging at any time, he will go free and receive a million bucks. Now the Judge doesn't want a lucky guess, so he gives himself 10 years in which to pick a day, so there's only a 1 /36502 ( two leap years) chance of a random guess. He then tells the man he is scheduled to be hanged October Fifth, 2015. Armed with his belief, that the date is unexpected, afterall, he said it was, the judge is astonished when the prisoner, with no hesitation, says, "Your Honor, I'm scheduled to die on October Fifth, 2015." The prisoner is released and given the million bucks.

Was this man Karnac, or is something else going on?

[t1mm01994]

Was this man Karnac, or is something else going on?

Do the exact same, but this time, dont tell him, and pick as date the 3rd to last day. The hangman is allowed on 1 day to say: This is the day on which I will be executed.

I wouldn't want to be in his shoes, that much is sure.

[billiams]

Was this man Karnac, or is something else going on?

Do the exact same, but this time, dont tell him, and pick as date the 3rd to last day. The hangman is allowed on 1 day to say: This is the day on which I will be executed.

I wouldn't want to be in his shoes, that much is sure.

There are plenty of ways to make it nearly impossible for the hanged man to predict the date. Just telling him to guess with no clues is one.

But my thought experiment was aimed at challenging Butler and Chapman's contention that existence of a contradiction generated by the conditions is in and of itself sufficient to render the students surprised, or the test unexpected in their parlance.

[Darrell88]

Still don't understand why this isn't classed simply as an inconsistent theory.

[jestingrabbit]

Because after the students prove it can't be on any day because they wouldn't be surprised, it then happens on a tuesday and they are surprised because they had proven that it couldn't happen then.

The professor is right about everything, so where's the inconsistency? In the analysis by the students? If that's where it is, you have to come up with some account of how it got there.

[billiams]

Still don't understand why this isn't classed simply as an inconsistent theory.

Darell, here is an example of a simple inconsistent set of premises that really would generate confusion: Prof. says, "The test is definitely tomorrow and, just so there is no misunderstanding the test will definitely not be tomorrow."

In this case, not using the assertive use of surprise to generate the inconsistency, students really would be surprised either way. I addressed this contingency a few posts above.

[billiams]

Because after the students prove it can't be on any day because they wouldn't be surprised, it then happens on a tuesday and they are surprised because they had proven that it couldn't happen then.

The professor is right about everything, so where's the inconsistency? In the analysis by the students? If that's where it is, you have to come up with some account of how it got there.

I agree with you and I don't agree with you.... Mainly, I do agree.

Please look at the Butler quote a few posts above and you'll see how the contradiction is teased out of the premises.

I think in our normal paralance, the Professor was indeed correct and no inconsistency obtained. It was when surprise , either covertly or overtly is redefined, that a contradiction can be generated. But even if it is, due to the linguistic features of surprise and it's underlying non- propositional nature when asserted, the inconsistency still doesn't change the students' knowledge vis a vis the test.

[Xias]

They could make a game show out of this.

You have a partner that you can't communicate with. There are ten numbered doors, one of which has a $1 million prize behind it. The timer begins, and one at a time the doors open (with some time in between). You have ten numbered buttons in front of you, and may press one (and only one) at any time before the corresponding door opens (so if you have not pressed a button by the time door 3 opens, you can press any number from 4 to 10.) You may not change your choice.

When the prize door opens, IFF both of you have selected that door, you split the prize. If one or both of you were wrong or didn't make a selection, you lose.

Before the game starts, the contestants are told that the prize has been placed so that there is no way for them to know with 100% certainty which door it is behind.

[billiams]

They could make a game show out of this.

You have a partner that you can't communicate with. There are ten numbered doors, one of which has a $1 million prize behind it. The timer begins, and one at a time the doors open (with some time in between). You have ten numbered buttons in front of you, and may press one (and only one) at any time before the corresponding door opens (so if you have not pressed a button by the time door 3 opens, you can press any number from 4 to 10.) You may not change your choice.

When the prize door opens, IFF both of you have selected that door, you split the prize. If one or both of you were wrong or didn't make a selection, you lose.

Before the game starts, the contestants are told that the prize has been placed so that there is no way for them to know with 100% certainty which door it is behind.

IFF they put it behind door 10, and I didn't win, I'd sue their asses!

But if all they mean is the more modest form of certainty that we actually use in real life, I might not win the lawsuit.

My analysis is this: the certainty thing adds nothing anyway, barring high redefinition (where it leads to... see above), so I would be unlikely to win the game.

I'm going to try to come up with a fun variation of the game where certainty can be used.

Change up the game a tad, where you have no paradox, but can actually use the certainty thing... same basic game, but there are 9 prizes 1 empty door, and you have have the condition that you can never know in advance (other than the empty door already being opened) which is the empty door, from door 9 till the last door is opened, and your job is to determine which door has a prize, and since the odds are so in your favor you have to do it a hundred times in a row, which door would you pick and why?

Seems like 10 becomes the lucky winner.

[Xias]

But if all they mean is the more modest form of certainty that we actually use in real life, I might not win.

My analysis is this: the certainty thing adds nothing anyway, barring high redefinition (where it leads to... see above), so I would be unlikely to win the game.

I'm going to try to come up with a fun variation of the game where certainty can be used.Change up the game a tad, where you have no paradox, but can actually use the certainty thing... same basic game, but there are 9 prizes 1 empty door, and you have have the condition that you can never know in advance (other than the empty door already being opened) which is the empty door, from door 9 till the last door is opened, and your job is to determine which door has a prize, and since the odds are so in your favor you have to do it a hundred times in a row, which door would you pick and why?

Seems like 10 becomes the lucky winner.

What's your point exactly? You took my game idea that is based on chance and made it a logically solvable game with a definite right answer. You turned the surprise exam paradox into the "Pick a day the exam is NOT on - and it can't be on Friday." exam paradox.

Look closely at my game proposal: What does the final line change about the game? Does them being told that "the winning door will be a surprise" change what door they might pick? Why? What information does that convey? Because if you say it proves that door 10 is impossible, then so are all of the doors. So then it is impossible to know what door will win, and therefore, it will be a surprise.

I think the issue is that the students think that the premise "On the day of the exam, you will be surprised" is somehow enough information to deduce which day it will be on, but it is not. As it has been said earlier in the thread, the deduction "the test cannot be given on friday" is based on the information "the test has not been given on monday, tuesday, wednesday, or thursday" which, on sunday night, they do not have. It's like saying "not A and not B and not C and not D implies E and F; but the professor said not F. Therefore, not E."

The reason why this is a paradox isn't in the same sense as "is the answer to this question 'no'?" is a paradox. It's more the counter-intuitive sort of paradox. The professor said it was a surprise, and it was - no paradox. But the student's logic seems legitimate and intuitive. Therefore, our intuition must be wrong. Where is it wrong? I would say that it is the step between "It can't be on Friday" and "It can't be on Thursday." On thursday before class, the students have not gotten the information "the test is not thursday" and so they can't say "the test must be Friday, therefore it is not a surprise, therefore it must be today."

What's interesting is, what is the difference between this line of nested thinking, and the line of nested thinking behind, say, the blue eyes problem?

[billiams]

But if all they mean is the more modest form of certainty that we actually use in real life, I might not win.

My analysis is this: the certainty thing adds nothing anyway, barring high redefinition (where it leads to... see above), so I would be unlikely to win the game.

I'm going to try to come up with a fun variation of the game where certainty can be used.

Change up the game a tad, where you have no paradox, but can actually use the certainty thing... same basic game, but there are 9 prizes 1 empty door, and you have have the condition that you can never know in advance (other than the empty door already being opened) which is the empty door, from door 9 till the last door is opened, and your job is to determine which door has a prize, and since the odds are so in your favor you have to do it a hundred times in a row, which door would you pick and why?

Seems like 10 becomes the lucky winner.

What's your point exactly? You took my game idea that is based on chance and made it a logically solvable game with a definite right answer. You turned the surprise exam paradox into the "Pick a day the exam is NOT on - and it can't be on Friday." exam paradox.

Look closely at my game proposal: What does the final line change about the game? Does them being told that "the winning door will be a surprise" change what door they might pick? Why? What information does that convey? Because if you say it proves that door 10 is impossible, then so are all of the doors. So then it is impossible to know what door will win, and therefore, it will be a surprise.

I think the issue is that the students think that the premise "On the day of the exam, you will be surprised" is somehow enough information to deduce which day it will be on, but it is not. As it has been said earlier in the thread, the deduction "the test cannot be given on friday" is based on the information "the test has not been given on monday, tuesday, wednesday, or thursday" which, on sunday night, they do not have. It's like saying "not A and not B and not C and not D implies E and F; but the professor said not F. Therefore, not E."

The reason why this is a paradox isn't in the same sense as "is the answer to this question 'no'?" is a paradox. It's more the counter-intuitive sort of paradox. The professor said it was a surprise, and it was - no paradox. But the student's logic seems legitimate and intuitive. Therefore, our intuition must be wrong. Where is it wrong? I would say that it is the step between "It can't be on Friday" and "It can't be on Thursday." On thursday before class, the students have not gotten the information "the test is not thursday" and so they can't say "the test must be Friday, therefore it is not a surprise, therefore it must be today."

What's interesting is, what is the difference between this line of nested thinking, and the line of nested thinking behind, say, the blue eyes problem?

Lighten up, my good man. I said there was no paradox in that game -- read what I wrote. I think I was clear. Also, I did not use the term in a way which highlights the second order predicate status of certainty, expected, and /or surprise . That's why I toyed with it. Just to see the difference in how it would look when the condition could be used. You really missed the content and tone of what I wrote.

As to impossibility, I addressed that above in detail in few posts just above. I'm surprised you didn't read them.... jeeees.

As to the description of the paradox, if certain uses ( not all I argue) of surprise, unexpected, et cetra are employed, the premises generate inconsistency between the conditions and their implication from which a contradiction can be derived. I also said in the same post you quoted, as well as all the others above, that the certainty condition added nothing to the students' information and would not allow them to deduce a day.

What I think we have here is a failure to communicate.

[Pingouin7]

The reasoning is bad because they can only suspect Friday IF THEY HAVEN'T HAD THE TEST YET. So pretend it's Tuesday and so far no exam. The days left to potentially house the exam are Wednesday, Thursday, and Friday.

IF they make it to Thursday AND there's no test, THEN they can suspect a test on Friday as it is the only day left in the week. Seeing as how they then expect the test, doing so will make it so that the professor wouldn't be keeping his word if he gave them a test and they can try to argue their way out of it if he still tries to give it.

The problem is that in order to suspect, and therefor invalidate Friday in the first place, you need to fulfill the two conditions of it currently being Thursday and there not having been a test yet.

Thus, they cannot suspect Thursday either by the same virtue. They need to already be suspecting Friday before Thursday becomes the effective end of the week and therefor also able to be suspected... but the thing is they can't suspect Friday until it's already Thursday!

So... I guess... ultimately the flaw arises in that the professor never said he wouldn't still try to give the exam on Friday to see if anybody would catch him.

But they know it cannot be on Friday, because if the exam is on Friday, they will know the exam is on that day before Friday happens.

[Loadstone]

What's interesting is, what is the difference between this line of nested thinking, and the line of nested thinking behind, say, the blue eyes problem?

Spoiler for referencing blue eyes solution:

Spoiler:

The blue eyes problem uses induction to eliminate possibilities upward (until a criteria is met, namely the known number of other blue-eyed inhabitants) so in no situation are all possibilities eliminated.

The surprise exam (or your related game) starts at the maximum possible value (Friday or Door #10) and eliminates it. It then uses induction to eliminate possibilities downward with no criteria to prevent all possibilities from simultaneous elimination.

Because after the students prove it can't be on any day because they wouldn't be surprised, it then happens on a tuesday and they are surprised because they had proven that it couldn't happen then.

The professor is right about everything, so where's the inconsistency? In the analysis by the students? If that's where it is, you have to come up with some account of how it got there.

The students' logic is incomplete. They must add another step to the end of their logic to reconsider any day(s) they "know" will not contain an exam to account for the stipulation that they will be surprised (and therefore can't know a day will not be the date of the exam). This final step in the logic might conclude that only these days are candidates for the exam. Since all days meet that requirement, the logic must be infinitely repeated, so no solution can be found.

If it were possible to logically conclude that only a single day could not contain an exam, then the students could then assume that it is indeed the exam date (assuming the students and professor are perfect logicians). Of course, if the professor also concludes this then you have a case of Wine in Front of Me (to borrow from Mafia terms).

[jestingrabbit]

I actually think the error is a little more fundamental than that Loadstone. I'm not claiming to have the problem entirely figured out, but I want to get this off my chest.

Firstly, surprise isn't a binary state. It might be surprising for the teacher to spring the test on tuesday, but it would be more surprising if on tuesday he turned the students into farm animals and had them reenact scenes from his favourite George Orwell novel, 1984. But then, the question becomes how to quantify surprise, and talk about how to maximise surprise (which is what the teacher wants) and minimise surprise (which is what the students want).

I propose 3 axioms that I'll show (or at least handwave) define surprise up to multiplication by a constant. So, for an event, E, let S(E) be a real number representing the quantity of surprise the event causes and I assert that:

1. S(E) is a function of the probability of E, P(E);
2. if P(A)<P(B), S(A)>S(B); and
3. if two events, A and B are independent ie P(A)P(B)= P(A and B), then S(A and B) = S(A) + S(B).

In 1 here, I mean probability in the Bayesian sense ie this is more an extended logic framework as outlined by ET Jaynes in "probability theory: the logic of science", rather than the more naive approach of only applying it to things like games of chance. We can use this sort of approach whenever we have some ignorance regarding the outcome of events, and I think the students are definitely in a state of ignorance regarding when the test is, and that surprise occurs when we are uncertain about things, that is, when our information is incomplete, or put another way, when we have some ignorance regarding the disposition of a system.

2 is simply the idea that rarer (or less believable) events are more surprising.

3 allows us to analyse sequences of events. For instance, if every week you buy a single raffle ticket and the same number of tickets are sold every week, then to win one week is half as surprising as to win two weeks in a row. Now, we could quibble about that, how exactly surprise works psychologically, but I feel like the algebraic advantages of this approach outweigh any error that it introduces. And anyway, this leads to some interesting results, so bear with me.

Now, you might remember 3 from the definition of logarithms. Not every function that obeys it is a logarithm, but every monotonic function (and axiom 2 guarantees monotonicity) with this property is a constant multiplied by a logarithm. The upshot of this is that we may let

S(E) = -log(P(E))

where the log is wrt some base, b>1 (and frankly I don't care which b we chose, they all just change the scale on our units of surprise (whether the unit of surprise is the wha, the huh or the gasp isn't something that is worth much thought imo)).

Now lets have T_i be the event "the test is on day i" (monday is day 1, tuesday day 2 etc) and we'll have p_i = "the belief/probability that the students have that the test will be on day i on night i-1, given that it hasn't happened yet".

Now, if all we want to do is minimise the surprise when the test happens, we could just say p_i=1 for all i, but that seems like a huge cheat. It implies that S(not T_i) = infinity, for instance. So, if those are the beliefs that you actually have, then your head would be exploding every day until the test happened, until you ended up looking all smug after the test actually happens, because you "knew" that it was going to happen on that day. This doesn't sit well with me.

So, what if we sum the test related surprise over the course of the week, what sort values for p_i reduce the possible surprise? If we assume that there is necessarily a test on one of the days, then we have p₅=1 ie it will have to happen on friday if it hasn't happened beforehand. This means that S(T₅) = 0 ie all the surprise happens earlier in the week.

Now consider the situation on thursday. Whatever surprise happens that day is the last surprise of the week. If S(T₄) != S(not T₄), then you've given the teacher a way to surprise you more or less depending on whether the test is on thursday or friday. So, you want there to be equality ie P(T₄) = P(not T₄) = 1 - P(T₄), so p₄ = 1/2, and the quantity of surprise that happens on thursday is -log(1/2) = log(2).

So, on wednesday, what you want is S(T₃) = S(not T₃) + log(2) ie the surprise that happens on wednesday is either the test, and its the last surprise of the week, or its the surprise from it not being the test, and you want that, plus the surprise from thursday, log(2), to be equal to the surprise from having the test. Solving, we have

-log(p₃) = -log(1- p₃) + log(2)
log(1- p₃) - log(p₃) = log(2)
log(1/p₃ - 1) = log(2)
1/p₃ - 1 = 2
p₃ = 1/3

and the amount of surprise that happens from wednesday on is log(3).

You can probably see where this is going, and you're right, you end up with p₂ = 1/4 and p₁ = 1/5. Put another way, the amount of surprise is minimised if, when there are n days left for the test to be on, you believe that the test will happen tomorrow with probability 1/n. What's more, even though I was just trying to minimise the maximum surprise, I also minimised the expected surprise.

The upshot of the calculations is that the students experience log(5) units of surprise, which is a little less surprised than they would be if they rolled a six the first time they rolled a dice. Its not a lot of surprise imo, but it is some.

Now, like I said at the start, I don't think this is the final and best analysis of the problem in the OP, but its an analysis that I like.

[Loadstone]

It's worth noting that, as stated, the problem indicates the surprise is binary since the students will either be surprised or won't be. This could be modified by saying the students would be "greatly surprised, where greatly implies a surprise value >= X".

My intuition bugs me about the conditional probabilities. From an minimal surprise standpoint, the surprise of the exam occurring on Monday S(T₁) should equal* the surprise of the exam not occurring on Monday S(T_>1). Therefore, the P(T₁) = P(T_>1) = 1/2, and S(T₁) = log(2).

The following are forecast probabilities (given what the students know initially, this is the distribution of likelihood of the exam on any given day).
So P(T₁) = P(T₂) + P(T₃) + P(T₄) + P(T₅) = 0.5.
P(T₂) must be equally likely to P(T_>2). So P(T₂) = P(T₃) + P(T₄) + P(T₅) = 0.25.
It follows that P(T₃) = 0.125 and P(T₄) = P(T₅) = 0.0625.

However, if on any given night the exam hasn't yet been given then the probabilities of prior days can be rolled proportionally into the remaining days. If the exam isn't given on Monday, then the conditional probably on night one (notation Pc_i where i represents how many days have passed) Pc₁(T₂) = 0.25 + 0.5 * P(T₁) = 0.5 since P(T₂) represents half of P(T_>1). Pc₁(T₃) = 0.125 + 0.25 * P(T₁) = 0.25 since P(T₂) represents one fourth of P(T_>1). It follows then that Pc₁(T₄) = Pc₁(T₅) = 0.125.

By extension:
Pc₂(T₃) = 0.5, Pc₂(T₄) = Pc₂(T₅) = 0.25.
Pc₃(T₄) = Pc₃(T₅) = 0.5.

So, on any given night the students can be equally sure that the exam will occur the next day and that it won't occur the next day. The only benefit I see to this approach is that it allows a prediction prior to the week that suggests 50% chance of exam by Monday, 75% chance by Tuesday, 87.5% chance by Wednesday, etc. Unfortunately, this merely tells the students that studying earlier is better, and since the students are notorious procrastinators this is ignored and nothing comes from the warning.



*I suggest this because surprise is minimized if you equally expect 2 mutually exclusive events to happen - an example would be if two sports teams are evenly matched, then the probability of each beating the other is 0.5 and whichever wins causes minimal surprise. However, if a team vastly outmatches the other with a probability of winning of 0.9 (other team has probability of winning 0.1) then the surprise of the inferior team winning is much greater.