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This "paradox" is no paradox at all, and has a ridiculously easy and obvious resolution. As a teacher: 1) you CAN guarantee for your students that there will

This "paradox" is no paradox at all, and has a ridiculously easy and obvious resolution. As a teacher: 1) you CAN guarantee for your students that there will be a pop quiz Monday through Friday of next week. 2) You CANNOT ALSO guarantee for them that they will be surprised that day. (i.e. that they won't be sure, on that day, that it's that day). This is because if it hasn't happened by the last day it can happen, it will happen the last day it can happen. This obviously makes sense. There's no paradox. There's nothing difficult. You simply cannot GUARANTEE them surprise, since out of whatever possible space of days it can happen, on the last day it won't be a surprise. Whether that space is 1, 2, 3, 4, or 5 days, on the last of that day it is not a surprise. For any space of days in which the exam can happen, the last day would not be a surprise; therefore, you cannot guarantee them both that there will be a space of days in which it can happen, and also GUARANTEE them that it will be a surprise. (in every eventuality. You can make them an 80% guarantee that it's a surprise, while making them a 100% guarantee that it's a quiz between monday and friday of next week. if you can increase the number of days you can increase your guarantee further and further. if I guarantee you a quiz on one of the next 100 days, I can also 99% guarantee you that you won't be sure you're having it that day -- i.e. will be 'surprised' -- on the day that I give it.) prior probability. pick a day monday to friday. quiz time comes. monday to thursday they're surprised; if it's on friday, they're not surprised. why you would think you could also surprise them after giving them a chance to "open" every day up to then is beyond me. THis is like giving you a guarantee: here are five envelopes, one has a heart in it. I guarantee you will be surprised when you open the one with a heart in it, even if you've opened four of the five envelopes already and found them empty. um....no... you can't guarantee that. it makes perfect sense, and there's no paradox or worth wasting any breath over.

tolmasky 15y ago
Your envelope example is great because it sheds even further light on the problem: the surprise does in fact exist, it just exists before the moment the final event takes place. In your envelope example, once you open the fourth envelope and discover nothing inside it you can simultaneously be surprised that it wasn't there and that the fifth envelope will contain the heart. In other words, the two events collapse into one. Similarly, when going to class on Thursday you will be "surprised" to find that the test is not that day but rather the next. The only thing that makes the exam version "confusing" is that you then have a full 24 hours to ruminate over this fact, thus making it seem like its not a surprise at all -- but its actually equivalent to on the first day being told when the test will be that week (at which point no one would rule out Friday) - it may be surprising at that moment but clearly won't be by the time the test actually takes place.
- its_so_on 15y ago
  right, exactly. This whole problem collapses if you don't 'open the envelopes one at a time' while having my 100% GUARANTEE that you won't be sure there's a heart in the one that does have one, after opening any others you've gone through. An 80% guarantee is fine, but 100% guarantee that you won't be sure, is incompatible with having a space of envelopes and going through it one by one. why would anyone think otherwise? the way in which I discovered this resolution is by coding up a perl script to 'monte carlo' different scenarios. I realized at the location that I made the teacher actually have to choose which day the exam will be, there is a space of days. (they have to choose - or end up choosing - as they break their first guarantee if they don't have it on any day of the week or more than one day of the week, or whatever. it has to be one and only one day of the week, however they end up getting there.) it doesn't matter if the teacher is choosing one of five days, 10 days, 100 days, whatever. The methodology is that the students get to go through the days one by one. If you are choosing one of five days, there is a 20% probability of choosing the last of them; therefore given this methodology you can only 80% guarantee them surprise. if you are choosing one of ten days, there is a 10% probability of choosing the last of them; therefore given the methodology you can only 90% guarantee them surprise if choosing one of 100 days, 1% probability you chose the last of them; you can 99% guarantee them surprise. Bottom line, which you discover if you code it up to run in simulations: at some point the teacher MUST actually choose a day to meet their first guarantee. (If they choose none - or end up giving it on none - then they've broken their first guarantee. Any algorithm that doesn't end up choosing a day 100% of the time is wrong). At the point of choosing a day, it doesn't matter if you make the Teacher choose monday through thursday, monday through wednesday, monday or tuesday, or hard-code it to choose Monday. Whatever the teacher chooses, if the students have access to the teacher's algorithm (which describes the space of possible days), then the last day of the space they would not have surprise. The teacher can thus give a guarantee equal to the number of days OTHER than the last day, over the number of days in the teacher's space. In the usual sense, this is 80% guarantee of surprise. of course, another question is if the teacher gets to follow an algorithm the students must only guess at. (They don't know what his algorithm is). In this case, for your simulation you can just hard-code the teacher always giving the test on Monday. Since the students don't know this is his algorithm, they will be "surprised" (they could have thought that he was picking from one day monday thru friday, they had no way to be sure that he was hard-coded to pick Monday), thus fulfilling both criteria of not being sure of what day the test is, and being given a test monday to friday. sonofabitch. new solution: The teacher can meet his obligations by being hard-coded to give the test on Monday, but not telling his students that this is the algorithm. Since they could be thinking he's choosing from a space of monday thru friday, they could think that the last day is friday: it's really Monday. this is an interesting aspect I hadn't considered (a secret algorithm). In this case I would say my response is more nuanced: 1) To whatever extent the students get to know of the teacher's algorithm for picking days, they are that much less able to be given a guarantee of 100% surprise on the day of the exam. Therefore, if the teacher is completely secret about his algorithm (i.e. he is NOT 'picking randomly monday thru friday') he can surprise the student. Any information he gives his students about his algorithm takes away from the extent to which he can guarantee their surprise. EVEN IF the algorithm includes randomness. In other words: if the students know the teacher is picking a random day monday thru friday, then he cannot offer a guarantee of surprising them. if the students have no idea what algorithm the teacher uses to pick a day (and, therefore, have no way to guess at the space), he can offer a guarantee of surprising them. Thus if students know nothing at all about the teacher, and he knows this fact, he can meet his obligation by giving the test from a space of one day (Monday) without any randomness or space of possibilities.
Symmetry 15y ago
You can't guarantee it for even simpler reasons than that - the students might be irrational. If, from a group of 5 students, one consults his tea leaves and is sure the exam will be on Monday, the next casts runes and is sure that the exam will be on Tuesday, etc then you can be sure at least one student won't be surprised.
- dextorious 15y ago
  Yeah, in the formal definition of the problem those kind of concerns do not matter...
its_so_on 15y ago
I see that this is getting upvoted. * Guys, I made a logical mistake here. I said "this is because if it hasn't happened by the last day it can happen"... But this supposes the students know what day it can happen. They might not. For example, the Teacher might give the test on Monday every year. Assuming the students have no contact with last year's students (or didn't bother to ask) they would not know that the space of possible days is Monday. Therefore, the teacher can make a guarantee that the test is on Monday through Friday, while saying that the students will be surprised. The students, thinking that the test may be Monday through Friday, would assume that the last day the test 'can happen' is Friday. But that's not a possibility at all. So this assumption would be a mistake. In fact, if the teacher is completely hiding his algorithm, then the students can be surprised. They don't know if the space of possible days is "only Monday", "only Tuesday", or "Monday, Tuesday, or Thursday", or whatever. Each of these spaces fulfills the requirement of some day monday thru friday next week: however, if the teacher keeps the space of possibilities quite secret, then the students cannot make their FIRST step of assumption, which is the last day that the thing "could" happen. So, I have two answers: 1) If the students know 100% of the teacher's ALGORITHM (how the teacher chooses days) -- even if it is random -- then the teacher cannot 100% guarantee them that they will be surprised. Namely, on the last day of the space the teacher chooses from, they would not be surprised. if, however, the teachers know NOTHING about the algorithm the teacher uses to determine the day, the teacher is indeed able to make his guarantee. In other words, if I am the teacher and I am trying to make this guarantee, I can do so, provided that instead of choosing a day randomly from monday through friday, I hard-code the day Monday, while keeping this algorithm secret from the students. TL;DR: if the teacher tells them only something ABOUT the day (that it is next week, and a monday or a tuesday or a wednesday or a thursday or a friday) then there is no paradox and he can guarantee that they will not know for sure of the day it is happening). If, however, they also know HOW the teacher chose the day (even if the algorithm includes randomness! i.e. 'a random day monday to friday') than the teacher may be unable to make his guarantee.

tolmasky 15y ago

Your envelope example is great because it sheds even further light on the problem: the surprise does in fact exist, it just exists before the moment the final event takes place. In your envelope example, once you open the fourth envelope and discover nothing inside it you can simultaneously be surprised that it wasn't there and that the fifth envelope will contain the heart. In other words, the two events collapse into one. Similarly, when going to class on Thursday you will be "surprised" to find that the test is not that day but rather the next. The only thing that makes the exam version "confusing" is that you then have a full 24 hours to ruminate over this fact, thus making it seem like its not a surprise at all -- but its actually equivalent to on the first day being told when the test will be that week (at which point no one would rule out Friday) - it may be surprising at that moment but clearly won't be by the time the test actually takes place.

its_so_on 15y ago
right, exactly. This whole problem collapses if you don't 'open the envelopes one at a time' while having my 100% GUARANTEE that you won't be sure there's a heart in the one that does have one, after opening any others you've gone through. An 80% guarantee is fine, but 100% guarantee that you won't be sure, is incompatible with having a space of envelopes and going through it one by one. why would anyone think otherwise? the way in which I discovered this resolution is by coding up a perl script to 'monte carlo' different scenarios. I realized at the location that I made the teacher actually have to choose which day the exam will be, there is a space of days. (they have to choose - or end up choosing - as they break their first guarantee if they don't have it on any day of the week or more than one day of the week, or whatever. it has to be one and only one day of the week, however they end up getting there.) it doesn't matter if the teacher is choosing one of five days, 10 days, 100 days, whatever. The methodology is that the students get to go through the days one by one. If you are choosing one of five days, there is a 20% probability of choosing the last of them; therefore given this methodology you can only 80% guarantee them surprise. if you are choosing one of ten days, there is a 10% probability of choosing the last of them; therefore given the methodology you can only 90% guarantee them surprise if choosing one of 100 days, 1% probability you chose the last of them; you can 99% guarantee them surprise. Bottom line, which you discover if you code it up to run in simulations: at some point the teacher MUST actually choose a day to meet their first guarantee. (If they choose none - or end up giving it on none - then they've broken their first guarantee. Any algorithm that doesn't end up choosing a day 100% of the time is wrong). At the point of choosing a day, it doesn't matter if you make the Teacher choose monday through thursday, monday through wednesday, monday or tuesday, or hard-code it to choose Monday. Whatever the teacher chooses, if the students have access to the teacher's algorithm (which describes the space of possible days), then the last day of the space they would not have surprise. The teacher can thus give a guarantee equal to the number of days OTHER than the last day, over the number of days in the teacher's space. In the usual sense, this is 80% guarantee of surprise. of course, another question is if the teacher gets to follow an algorithm the students must only guess at. (They don't know what his algorithm is). In this case, for your simulation you can just hard-code the teacher always giving the test on Monday. Since the students don't know this is his algorithm, they will be "surprised" (they could have thought that he was picking from one day monday thru friday, they had no way to be sure that he was hard-coded to pick Monday), thus fulfilling both criteria of not being sure of what day the test is, and being given a test monday to friday. sonofabitch. new solution: The teacher can meet his obligations by being hard-coded to give the test on Monday, but not telling his students that this is the algorithm. Since they could be thinking he's choosing from a space of monday thru friday, they could think that the last day is friday: it's really Monday. this is an interesting aspect I hadn't considered (a secret algorithm). In this case I would say my response is more nuanced: 1) To whatever extent the students get to know of the teacher's algorithm for picking days, they are that much less able to be given a guarantee of 100% surprise on the day of the exam. Therefore, if the teacher is completely secret about his algorithm (i.e. he is NOT 'picking randomly monday thru friday') he can surprise the student. Any information he gives his students about his algorithm takes away from the extent to which he can guarantee their surprise. EVEN IF the algorithm includes randomness. In other words: if the students know the teacher is picking a random day monday thru friday, then he cannot offer a guarantee of surprising them. if the students have no idea what algorithm the teacher uses to pick a day (and, therefore, have no way to guess at the space), he can offer a guarantee of surprising them. Thus if students know nothing at all about the teacher, and he knows this fact, he can meet his obligation by giving the test from a space of one day (Monday) without any randomness or space of possibilities.

Symmetry 15y ago

You can't guarantee it for even simpler reasons than that - the students might be irrational. If, from a group of 5 students, one consults his tea leaves and is sure the exam will be on Monday, the next casts runes and is sure that the exam will be on Tuesday, etc then you can be sure at least one student won't be surprised.

dextorious 15y ago
Yeah, in the formal definition of the problem those kind of concerns do not matter...

its_so_on 15y ago

I see that this is getting upvoted. * Guys, I made a logical mistake here. I said "this is because if it hasn't happened by the last day it can happen"... But this supposes the students know what day it can happen. They might not. For example, the Teacher might give the test on Monday every year. Assuming the students have no contact with last year's students (or didn't bother to ask) they would not know that the space of possible days is Monday. Therefore, the teacher can make a guarantee that the test is on Monday through Friday, while saying that the students will be surprised. The students, thinking that the test may be Monday through Friday, would assume that the last day the test 'can happen' is Friday. But that's not a possibility at all. So this assumption would be a mistake. In fact, if the teacher is completely hiding his algorithm, then the students can be surprised. They don't know if the space of possible days is "only Monday", "only Tuesday", or "Monday, Tuesday, or Thursday", or whatever. Each of these spaces fulfills the requirement of some day monday thru friday next week: however, if the teacher keeps the space of possibilities quite secret, then the students cannot make their FIRST step of assumption, which is the last day that the thing "could" happen. So, I have two answers: 1) If the students know 100% of the teacher's ALGORITHM (how the teacher chooses days) -- even if it is random -- then the teacher cannot 100% guarantee them that they will be surprised. Namely, on the last day of the space the teacher chooses from, they would not be surprised. if, however, the teachers know NOTHING about the algorithm the teacher uses to determine the day, the teacher is indeed able to make his guarantee. In other words, if I am the teacher and I am trying to make this guarantee, I can do so, provided that instead of choosing a day randomly from monday through friday, I hard-code the day Monday, while keeping this algorithm secret from the students. TL;DR: if the teacher tells them only something ABOUT the day (that it is next week, and a monday or a tuesday or a wednesday or a thursday or a friday) then there is no paradox and he can guarantee that they will not know for sure of the day it is happening). If, however, they also know HOW the teacher chose the day (even if the algorithm includes randomness! i.e. 'a random day monday to friday') than the teacher may be unable to make his guarantee.