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please see my other comment. Your formulation (1) is wrong, and it matters. If x is, as you say "a random variable from the set {1,2,3,4,5}, with unknown dist

please see my other comment. Your formulation (1) is wrong, and it matters. If x is, as you say "a random variable from the set {1,2,3,4,5}, with unknown distribution" then it is true that you cannot, given the methodology of going thru days one by one, be 100% guaranteed to not be able to deduce the day of the test that you must have it that day. This is because for whatever space of possibilities (in this case 1,2,3,4,5) on the last day of the space, you can deduce (given the methodology) that it must be that day. But the teacher is not really giving you a gurantee that x is a random variable from the set {1,2,3,4,5}. the teacher is only giving you a guarantee that x is a member of the set 1,2,3,4,5. You have no idea what x is, nor how the teacher chose it. So, given the guarantees Version A 1) x is a random variable from the set (1,2,3,4,5) 2) prior to the first day (commencement) you are given a 100% guarantee that proceeding thru opening days one by one until day x, you will not be able to deduce on day x, before opening that day, that the exam will be given that day. in this case you are right: there can be no solution. but version b is different: 1) teacher guarantees x is 1 or 2 or 3 or 4 or 5. 2) prior to the first day (commencement) teacher gives 100% guarantee that proceeding thru opening days one by one until day x, you will not be able to deduce on day x, before opening that day, that the exam will be given that day. in this case you don't really know what set(s) x is a member of. The teacher could fulfill his obligation for (2) by hard-coding x to be Monday. The student could reason that "If by Friday we don't have an exam, the exam must be Friday", thinking that the space is 1,2,3,4,5. But it's not: there is no possibility the exam is on Friday, due to hard-coding the exam to be Monday, and therefore Friday is not the 'last day' the exam could be given. It's not any of the days the exam could be given. Nor is Thursday. Nor is Wednesday. Therefore, the student is simply wrong in supposing (guessing) that this is the space in which the exam is on one of the days of.

ajuc 15y ago
> But the teacher is not really giving you a gurantee that x is a random variable from the set {1,2,3,4,5}. Why? Even if X is always 1, it is a random variable from {1,2,3,4,5}, with distribution {1.0, 0.0, 0.0, 0.0, 0.0}, which I don't know so it is unknown for me. For me teachers method of choice is not important - he can throw the dices, he can set it to 1. For me it's still random variable, because I don't know the result. If the real space is {1,2}, because teacher will throw a coin, I can still think about it as variable on {1,2,3,4,5}, because it's all the same, when the probabilities are {0.5, 0.5, 0, 0, 0}.
- its_so_on 15y ago
  ajuc: thank you for your answer. Let's look through this in pseudocode. #teacher picks day DayOfTest = "Monday"; #now go through the week. for today in (Monday, Tuesday, Wednesday, Thursday, Friday): print "It's ", today, " today." if student_is_sure_test_today(today) == true: if today == DayOfTest: print "PARADOX!! TEACHER GUARANTEED SURPRISE but student knew test today.." else: print "STUDENT ERROR!! Was 'sure' test today, but there is no test today." else: if today == DayOfTest: print "TEST today. Student surprised. No paradox." exit else: print "Day ", today, " over without a test." Now you run this program 1,000,000 times. If instead of DayOfTest = Monday we made it random monday through friday, then on the Fridays, the student would know it's that day, and we would trip the PARADOX!!! error message. In other words, we would only have an 80% guarantee. In 20% of cases we would trip a paradox. (since STUDENT ERROR!! is never tripped, it's a true paradox and not a mistaken student algorithm.). But if you run the pseudocode as-is above a million times, the student would NEVER get to say, on a Friday, that they are sure it's that day. You would not print a single PARADOX!!!! error message. Because you don't print a single paradox, we don't have to remove that day (reducing the possible spaces from five to four) to avoid it. Do you see? If the pseudocode has the teacher giving the day as random monday through friday, then he can only give an 80% guarantee of not tripping a paradox. but if the teacher can have a secret algorithm (giving the test on Monday) then the student's very first line of reasoning fails: they do not get to trip any paradox error messages or, in order to remove these error messages, remove any possible days from the space.
  - ajuc 15y ago
    We differ in interpretations. For me, even if student correctly guessed, that the test will be on Monday, if he couldn't deduce this from the assumptions, he will be surprised. Like when you roll the dice - you can guess that the dice will roll 6, sometimes you'll be right, but you'll still be surprised if it rolls 6. If someone told you before the roll, that this dice always rolls 6 - you'll know that, and then you won't be surprised. The dice is the same, your guess is the same, what changed is your knowledge. In our example teacher provided us with knowledge that is contradictory. You can deduce anything you want from 0=1, so the whole exercise is futile.

ajuc 15y ago

> But the teacher is not really giving you a gurantee that x is a random variable from the set {1,2,3,4,5}. Why? Even if X is always 1, it is a random variable from {1,2,3,4,5}, with distribution {1.0, 0.0, 0.0, 0.0, 0.0}, which I don't know so it is unknown for me. For me teachers method of choice is not important - he can throw the dices, he can set it to 1. For me it's still random variable, because I don't know the result. If the real space is {1,2}, because teacher will throw a coin, I can still think about it as variable on {1,2,3,4,5}, because it's all the same, when the probabilities are {0.5, 0.5, 0, 0, 0}.

its_so_on 15y ago
ajuc: thank you for your answer. Let's look through this in pseudocode. #teacher picks day DayOfTest = "Monday"; #now go through the week. for today in (Monday, Tuesday, Wednesday, Thursday, Friday): print "It's ", today, " today." if student_is_sure_test_today(today) == true: if today == DayOfTest: print "PARADOX!! TEACHER GUARANTEED SURPRISE but student knew test today.." else: print "STUDENT ERROR!! Was 'sure' test today, but there is no test today." else: if today == DayOfTest: print "TEST today. Student surprised. No paradox." exit else: print "Day ", today, " over without a test." Now you run this program 1,000,000 times. If instead of DayOfTest = Monday we made it random monday through friday, then on the Fridays, the student would know it's that day, and we would trip the PARADOX!!! error message. In other words, we would only have an 80% guarantee. In 20% of cases we would trip a paradox. (since STUDENT ERROR!! is never tripped, it's a true paradox and not a mistaken student algorithm.). But if you run the pseudocode as-is above a million times, the student would NEVER get to say, on a Friday, that they are sure it's that day. You would not print a single PARADOX!!!! error message. Because you don't print a single paradox, we don't have to remove that day (reducing the possible spaces from five to four) to avoid it. Do you see? If the pseudocode has the teacher giving the day as random monday through friday, then he can only give an 80% guarantee of not tripping a paradox. but if the teacher can have a secret algorithm (giving the test on Monday) then the student's very first line of reasoning fails: they do not get to trip any paradox error messages or, in order to remove these error messages, remove any possible days from the space.
- ajuc 15y ago
  We differ in interpretations. For me, even if student correctly guessed, that the test will be on Monday, if he couldn't deduce this from the assumptions, he will be surprised. Like when you roll the dice - you can guess that the dice will roll 6, sometimes you'll be right, but you'll still be surprised if it rolls 6. If someone told you before the roll, that this dice always rolls 6 - you'll know that, and then you won't be surprised. The dice is the same, your guess is the same, what changed is your knowledge. In our example teacher provided us with knowledge that is contradictory. You can deduce anything you want from 0=1, so the whole exercise is futile.