The Surprise Exam Paradox, Part 2

The Surprise Exam Paradox (which is equivalent to the Unexpected Hanging Paradox) is a mind-boggling logical conundrum. In Part 1, we ran through our own variation of this paradox. Here in Part 2, we give an in-depth analysis of the paradox - and attempt to resolve it.

[iyqz]

This paradox is really hard to think through, Mr. Harris should just pretend to fall into Alice's logic trap but split the exam into multiple parts and give it out throughout the week. The students would definitely be surprised. The surprise element is then the format of the exam (parts) and not the days itself.

[HenryLoenwind]

This only solves the paradox if we accept that there's a 20% chance that the teacher's announcement is false. A Friday announcement (i.e. 20% of the possible days) cannot be a surprise, and as such, it would violate the stated rules. So if we assume on Wednesday evening that the teacher will follow the stated rules, this rules out Friday and leaves only one possible day---which in turn wouldn't be a surprise either. The only way to resolve this is to fix the contradiction in the announcement.

This very much sounds like fighting users about software requirements stated in contradictory absolutes.

[gregorymoore]

Beautiful content, nicely presented. I must admit that halfway through your elimination of the possible attitudes, I almost expected you to start saying that you could clearly not choose the wine in front of you. 😉

[InkFPS]

What of the scenario where students study beforehand and are prepared for an exam any day of the week? Whether it is a surprise or not is no longer relevant because they were told there would be a 'surprise' exam, meaning It isn't much of a surprise regardless of when it occurs.

[PowerStar]

There is one key element that is being left out:

Mr. Harris must promise that there will be *exactly 1* exam next week.

Otherwise there could for example, be an exam on Monday and then also an exam on Friday, both of which are a surprise.

[JacobMutuku-mz]

I think the flaw is grounded in the elimination process. According to her the initial day to be eliminated is Friday which makes sense, however, on Wednesday evening she doesn't consider Friday as a possible day at least for a non-surprise exam. In this way she trims the week into having only 4 possible days hence making her conclusion of rulling out Thursday for a surprise exam true. Am I even making sense myself?

[KaiHenningsen]

I don't think your solution fits your constraints. I think if you accept the logic that under the given conditions, the last day is impossible, then the same logic will always eliminate the last day. I think there's something wrong with shrinking the interval by cutting out the last day, but I'm not sure what it is. But once you accept that shrinking, I think the rest must follow.

[orenzeshani]

Every time a military surprise attack succeeds, this is what happened

[justgivemethetruth]

Can the students only be surprised at the time of the exam? Seems like if the test is given on Friday the surprise occurs when they get the to into Thursday and there is no exam, so the surprise is that the test is given on Friday, but it is revealed on Thursday's class? ;-)

[sleepingdragon]

I am not a logician but I think the error is in allowing Alice to work backwards through time - the things you come to know on Thursday are not yet known on Wednesday, and so on. So while it is true that you cannot be surprised on Thursday evening if no exam has occurred (p=1 that it will be Friday) this is not true on Wednesday (p=.5) or Tuesday (p=.3) or Monday (p=.25) or Sunday (p=.2) - where p = probability that the exam will be tomorrow.

[ForsmanTomas]

What he should have said was 'The earliest point you can know what day the exam will be is at the end of school on Thursday.' And if he wants to be cheeky about it he can add 'I might decide on a day by the roll of a dice, or I might have already decided.'
The fault was not in her logic but in the definition of surprise and with it giving himself the limitation of having the surprise having to be be so close to the test being handed out. If his definition is true, she is correct and using the uncertainty that he might be wrong is a cop out and not a solution to the logical paradox.

[ABackZhu]

I think the problem is the term "next week". Once the teacher defines that the surprising exam is in the "next week", it is not a surprise anymore. Limiting the time period to "next week" eliminates the possibility to have continuous surprises after the last day (Friday) of "next week". Defining "next week" has the same effect as defining "next day". If the teacher says he gives a surprising exam "next day", it is not a surprising exam anymore for a time unit of a "day" unless we further keep that surprise in hours, minutes or even seconds.

[toutenunmot]

If you decouple the surprise happening from the unveiling then you resolve the paradox. If the exam happens Monday, Tuesday, Wednesday or Thursday, the teacher can say: surprise the exam is now. If the exam is Friday, the teacher can say on Thursday: surprise the exam is tomorrow. I feel like it qualifies as a surprise exam.

[KarlWork-ni]

The way to understand the problem is to realize TIME is involved and the information you have as you progress thru the week changes.

So the teacher makes the announcement before the week begins. You cannot use this BACKWARDS REASONING before week begins because you don't yet know the exam has not taken place on Monday or Tuesday etc. That BACKWARDS REASONING would involve using information you don't have!!

On Tuesday evening you know exam didn't happen Monday so if it takes place on Wednesday it would be a surprise.

Clearly on Thursday evening you know it must take place Friday because you NOW KNOW it's not yet taken place so it's not going to be a surprise.
There is no paradox YOU CANNOT USE THIS BACKWARDS REASONING BECAUSE YOU WILL BE USING INFORMATION YOU DON'T HAVE.

You cannot (on Sunday or on Monday or on Tuesday etc) reason it will not happen Friday because you don't yet know if it happened on those Monday or Tuesday etc.

So in summery teacher makes announcement on Sunday and you start using this reasoning to work out cannot be Friday YOU CANNOT because you are using information you don't
The exam could be the following day Monday. You cannot use this backwards reasoning because it involves declaring exam didn't happen Monday BUT YOU DON'T YET KNOW WHETHER IT HAPPENED MONDAY.

So you cannot use this backwards reasoning because it ALWAYS involves you using information that YOU DON'T

SORTED!

[KarlWork-ni]

Another way to think about it is this. The rules of this Surprise Exam must be made very clear on the Sunday.
If it's the usual Monday to Friday and will not know evening before then that is NOT as student points

You can still have a surprise exam tho rules would have to be different because if it doesn't happen by Thursday you know it's going to be Friday. That's how it happens in real life if you like.

The original surprise exam is NOT and I think everyone was assuming it was, there is no paradox.

[KarlWork-ni]

If on Sunday student says "it cannot take place Friday" you ask WHY?
Student says "if it didn't take place Monday etc etc etc etc"

IF IF IF IF (IT'S STILL ONLY

Cannot use reasoning like that.

[oximas-oevf]

well ya, a Thursday exam would be a surprise, as on Wednsday evening there are still two days left

when you say that it's impossible for a surprise exam to be on Friday that's simply not true
as it's only impossible if we are on Thursday evening not on Wednesday evening.

As such a Thursday exam would be a surprise,
similarly it follows that an exam on any of the days would also be a surprise.

from what I understood, the flow is in saying it's "IMPOSSIBLE" for the exam to be on Friday, the fact is it's not "impossible"
unless you are on Tuesday evening (that is assuming an exam must take place of course).

[thorstenl.]

You could do some tasks on thursday and reveal on friday, that these tasks from thursday will be count as an exam…. Surprise… you already did it 😂

[KarlWork-ni]

So the error is in the first few seconds of video, the student immediate reply to teacher. Think about it - when she says what she does SHE CANNOT BECAUSE SHE IS USING INFORMATION SHE DOESN'T YET

[johnvriezen]

Lets break it down: The teacher makes claim E: "There will be an exam next week". and also claim S: "There is a 100% chance the day of the exam will not be known to you until the day I give the exam."

Lets break it down further: The teacher claims: ((EMon or ETue or EWed or EThu or EFri) and S). Combining this with the assumptions about how the passage of time works and so forth, Alice concludes (not EFri) then concludes (not EThu) etc. until she concludes (not EMon) and hence: (Not E). This makes the original claims self contradictory (much like the statement "This statement is a lie".) It just takes longer to get to the contradiction. When our premises lead to a conclusion that is a contradiction, we don't just assume the conclusion is true, we assume one or more of the premises are wrong.

So either E is false or S is false or both are false. If E is false, S is not even a statement that has a truth value. If S is false then E can be true without contradiction. However (not S) is the statement: "There is NOT a 100% chance the day of the exam will not be known to you until the day I give the exam." In fact, if the exam is Friday it will not be a surprise. If it is any other day, it will be a surprise. That's how the students were surprised on Wednesday.


A totally different way of looking at it: What is a surprise? A surprise is when something happens (exam happening on a particular day) when you were reasonably confident it would not happen or had no knowledge of whether it would happen. Alice was 100% sure the exam would not be on Friday (or any other day). She should have then concluded that she would be surprised if the exam actually did occur on ANY of those days.