4 ms·

couldn't they just say, assuming iid: on Thursday you know that the probability that the exam will be on Friday is 1 on Wednesday you know that the probabi

by DrJ 15y ago

couldn't they just say, assuming iid: on Thursday you know that the probability that the exam will be on Friday is 1 on Wednesday you know that the probability that the exam will occur Thursday is .5 etc. etc. on Sunday you know that the probability that the exam will occur on Monday is .2 So if you look at it from (naive) probability, you can never make the 2nd assumption that on Thursday you will not have surprise test. you could say the pmf is: .2, .5 .8 .17 .5 and the cdf is .2 .25 .33 .5 1 Which from memory looks like a Beta Distribution.

Retric 15y ago
The problem with this question is it's simply a logical contradiction. If I say a = 1, b = 2, a = b most people will object. But, simply saying a = 1, b = 2, c = b, d = c... z = a then most people give up on that conviction and assume they made a mistake. PS: I suspect this comes from the limits of logic in the real world. If I do the same thing with a really long computer program it might though some internal errors flip that 2 into a 1 even as you simply copy it from one memory location to another.
r00fus 15y ago
In the article, the students' logic is based on semantics. "Surprise" should really be "Random". Ie, the surprise is a one time event (i.e., "Random day next week - surprise!"). So as of the moment of notice there is a random choice, and as that day approaches, it becomes less of an ongoing "surprise".
prof_hobart 15y ago
No they couldn't. If the statement was "the exam is going to be on a randomly selected day next week", then you'd be correct. But that's not the case. The teacher claims that "students won’t know exactly which day until the exams are handed out", whereas by the end of Thursday, they would be 100% certain that the exam was going to be on Friday. Therefore it is impossible for a surprise exam to be on the Friday, so that can be ruled out as a possibility. Which means that by the end of Wednesday, the only possible day for a surprise exam becomes Thursday, and the same logic which ruled out Friday then applies to Thursday (if it has to be Thursday, it can't be a surprise, so it can't be Thursday), and all the way back through the rest of the week.
- pierrebai 15y ago
  There a few traps in the question, leading to different refutation or explanation. The first is the ambiguous definition of surprise. This lead to easy explanation of the paradox. The teacher might as well hand out an exam on a red piece or paper and claim that that is a surprise. The second is that the teacher invalidly presumes to be able to predict the mental state of his students. This can be stated simply: over the weekend, all his students die unexpectedly. The following week, none of his student are surprised since the set of surprised students is empty. We can work around those two flaw by ignoring the no-student case and assuming that "surprised" is replaced by a more exact formulation: "on the day the exam occurs, nobody would be able to say ``I knew the exam was today''". In this formulation, I contend that nothing can be concluded. The reason for this is that induction can only be applied on facts. The future is not a fact, so to apply induction on a sequence of future events is an error. The only conclusion that one can reach is the contrafactual: if we were Friday then the exam cannot occur on Friday. This is because the reasoning does not rely on any inductive argument but rely only on things that would be facts inside the contrafactual: if we were on Friday now, it would be false that the exam cannot be held on that day and that nobody could say that the exam is to be held that day. A contrafactual of this nature cannot be built for Thursday, since in that contrafactual ("if we were Thursday...") one would have to reason by induction about teh future. Of course, in the Thursday-contrafactual, one could state the Friday-contrafactual, but you can't build inudction on contrafactual (i.e. non-fact).
  - prof_hobart 15y ago
    >The first is the ambiguous definition of surprise. He doesn't actually use the word "surprise" anywhere in his statement, and spells out pretty clearly what the nature of the surprise would be - "students won’t know exactly which day until the exams are handed out. Nothing, other than coming in on a particular day, not knowing for certain that the exam will be that day, but getting the exam on that day nevertheless, would do - e.g.an exam on the Friday would not qualify, as you would already know that this was the day of the exam. And nor would red paper. >The following week, none of his student are surprised since the set of surprised students is empty. Again, he doesn't claim that anyone would be surprised. He said that they "won’t know exactly which day until the exams are handed out." I suppose you could argue that if they are dead, then they won't even know on that day either, but that's a pretty tenuous point and anyway technically he doesn't explicitly claim that they would know at that point, just that they wouldn't beforehand. >"on the day the exam occurs, nobody would be able to say ``I knew the exam was today''". Which is pretty much what he did say. >A contrafactual of this nature cannot be built for Thursday Why not? The argument against it being on Friday (and I think we're both agreed that it's impossible for it to be on Friday and still line up with his statement) is that if there's only one possible choice for what day it's on, then the pupils - or at least any that are still alive - would know that the exam was on that day. And if it can't be on Friday, then when it gets to first thing Thursday how many choices are left for possible days for the exam to line up with his statement? One - so we're back to it being a certainty as to which day it would be.

by DrJ 15y ago

Retric 15y ago

The problem with this question is it's simply a logical contradiction. If I say a = 1, b = 2, a = b most people will object. But, simply saying a = 1, b = 2, c = b, d = c... z = a then most people give up on that conviction and assume they made a mistake. PS: I suspect this comes from the limits of logic in the real world. If I do the same thing with a really long computer program it might though some internal errors flip that 2 into a 1 even as you simply copy it from one memory location to another.

r00fus 15y ago

In the article, the students' logic is based on semantics. "Surprise" should really be "Random". Ie, the surprise is a one time event (i.e., "Random day next week - surprise!"). So as of the moment of notice there is a random choice, and as that day approaches, it becomes less of an ongoing "surprise".

prof_hobart 15y ago

No they couldn't. If the statement was "the exam is going to be on a randomly selected day next week", then you'd be correct. But that's not the case. The teacher claims that "students won’t know exactly which day until the exams are handed out", whereas by the end of Thursday, they would be 100% certain that the exam was going to be on Friday. Therefore it is impossible for a surprise exam to be on the Friday, so that can be ruled out as a possibility. Which means that by the end of Wednesday, the only possible day for a surprise exam becomes Thursday, and the same logic which ruled out Friday then applies to Thursday (if it has to be Thursday, it can't be a surprise, so it can't be Thursday), and all the way back through the rest of the week.

pierrebai 15y ago
There a few traps in the question, leading to different refutation or explanation. The first is the ambiguous definition of surprise. This lead to easy explanation of the paradox. The teacher might as well hand out an exam on a red piece or paper and claim that that is a surprise. The second is that the teacher invalidly presumes to be able to predict the mental state of his students. This can be stated simply: over the weekend, all his students die unexpectedly. The following week, none of his student are surprised since the set of surprised students is empty. We can work around those two flaw by ignoring the no-student case and assuming that "surprised" is replaced by a more exact formulation: "on the day the exam occurs, nobody would be able to say ``I knew the exam was today''". In this formulation, I contend that nothing can be concluded. The reason for this is that induction can only be applied on facts. The future is not a fact, so to apply induction on a sequence of future events is an error. The only conclusion that one can reach is the contrafactual: if we were Friday then the exam cannot occur on Friday. This is because the reasoning does not rely on any inductive argument but rely only on things that would be facts inside the contrafactual: if we were on Friday now, it would be false that the exam cannot be held on that day and that nobody could say that the exam is to be held that day. A contrafactual of this nature cannot be built for Thursday, since in that contrafactual ("if we were Thursday...") one would have to reason by induction about teh future. Of course, in the Thursday-contrafactual, one could state the Friday-contrafactual, but you can't build inudction on contrafactual (i.e. non-fact).
- prof_hobart 15y ago
  >The first is the ambiguous definition of surprise. He doesn't actually use the word "surprise" anywhere in his statement, and spells out pretty clearly what the nature of the surprise would be - "students won’t know exactly which day until the exams are handed out. Nothing, other than coming in on a particular day, not knowing for certain that the exam will be that day, but getting the exam on that day nevertheless, would do - e.g.an exam on the Friday would not qualify, as you would already know that this was the day of the exam. And nor would red paper. >The following week, none of his student are surprised since the set of surprised students is empty. Again, he doesn't claim that anyone would be surprised. He said that they "won’t know exactly which day until the exams are handed out." I suppose you could argue that if they are dead, then they won't even know on that day either, but that's a pretty tenuous point and anyway technically he doesn't explicitly claim that they would know at that point, just that they wouldn't beforehand. >"on the day the exam occurs, nobody would be able to say ``I knew the exam was today''". Which is pretty much what he did say. >A contrafactual of this nature cannot be built for Thursday Why not? The argument against it being on Friday (and I think we're both agreed that it's impossible for it to be on Friday and still line up with his statement) is that if there's only one possible choice for what day it's on, then the pupils - or at least any that are still alive - would know that the exam was on that day. And if it can't be on Friday, then when it gets to first thing Thursday how many choices are left for possible days for the exam to line up with his statement? One - so we're back to it being a certainty as to which day it would be.