3 ms·

You say: "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}" But IF, insofar as "x is alwa

You say: "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}" But IF, insofar as "x is always monday", x is a 'random variable from monday,tuesday,wednesday,thursday,friday' with distribution{1.0,0,0,0,0} THEN, it is just as true to say insofar as "x is always monday", x is a 'random variable from monday,tuesday,wednesday,thursday,friday,not given at all' with distribution{1.0,0,0,0,0,0} isn't it, though? That doesn't mean there's a "chance" that it's 'not given at all'. The teacher gives you a guarantee that "not given at all" is 0.0 (by saying that it is either monday, tuesday, wednesday,thursday, or friday, and will actually be given) -- he doesn't give you a guarantee that any specific days are nonzero. In the end, "not given at all" COULD BE, depending on the algorithm the teacher uses to select days, JUST AS likely as Friday. That is, if Friday has a 0.0 chance of happening. Intuitively, it sounds like we are given some guarantee that Friday is more likely than No Test Given At All. In fact, we are given no such guarantee. Insofar as we have such a guarantee, the teacher cannot guarantee surprise. It's quite simple, really.

ajuc 15y ago
I didn't assumed Friday probabilty is > 0, nor any other day. I didn't assumed X is not random variable with other options. I just assumed all the other options for sure have probability of 0, so can be ommited, and for days 1..5 I didn't assumed anything other than teacher guaranted. But we are given guarantee that no day will have probability 1.0, if I know it wasn't any day before. From the guarantee we know it won't be Friday. From this we know it won't be Thursday, and so on. From the fact that teacher told us that there will be an exam this week, we know any other options we could add (like "not given at all", or "next monday") have probability of 0. From this all we see that teacher claims were contradictory.
- its_so_on 15y ago
  Right, you are given the guarantee: "that no day will have probability 1.0, if I know it wasn't any day before". The key is that "if I know it wasn't any day before" CAN'T be a false premise. If the teacher secretly fixes the day at Monday, then the first line of reasoning becomes FALSE. The student is WRONG to argue "If I make it to Friday..." because that's not possible any more than "If I make it to Saturday..." or "If I make it to next week..." All of these are false premises. Arguments from a false premise don't lead to any conclusions. "If your mother is named John, then..." is a mistaken argument, if there is a 0.0 chance your mother is named John (for some reason). It doesn't matter what the other half of the sentence is, if that premise has a 0.0 chance of being true, you can't really deduce anything from it (if it has 0.0 chance of being true). The thing that makes this thing hard is you don't KNOW how the teacher chooses the day. If the teacher always picks Monday, but nobody in the class knows it, they don't know that Friday has a 0.0 chance of happening, they begin to reason "If we make it to Friday morning without a test..." and this argument is simply FALSE. It's about something not possible. It is like saying "If 1 = 2..." The student makes a mistake (if the teacher always gives a test Monday) by saying "If we make it to Friday without a test..." just as the student makes a mistake if they finish the sentence "If 1 = 2..." I mean, I can continue making arguments: "If 1 = 2, then 2 = 4". Is that a "true" statement? Not really. You can't really deduce anything about 2 being equal to 4 or not if 1 can't equal 2. You can't really deduce anything about other days of the week, if it's not possible for the teacher to pick Friday. That is why it matters how the teacher chooses the days.
  - ajuc 15y ago
    Nitpicking: "If 1 = 2, then 2 = 4" technically this is a true statement. Any implication starting with false premise is true. But you already know that, and you meant that I cannot use the "then" clause of this implication, because it can be true or false. Main subject: My reasoning resulted with contradiction, which I blamed on assumption (1) or (2), but it can be that there was hidden assumption (3) "there is a chance that Friday will be choosen". When we assume (1) (2) and (3) we have contradiction, when we assume (1) and (2), but not (3), we can't use (2) to prove there's no possible day to choose for teacher, so that case is the solution. OK - count me persuaded. Thanks for disscussion.
  - prof_hobart 15y ago
    >"If we make it to Friday morning without a test..." and this argument is simply FALSE. It's about something not possible. It is like saying "If 1 = 2..." Well, yes it is like saying "if 1=2, then 2=4", but that doesn't mean that this line of thinking is an invalid one. It simply means that by showing the result to be wrong, the test at the start is also wrong. The statement isn't attempting to prove that 2=4 by assuming that 1=2. It's saying that we can show that 1 quite clearly doesn't equal 2 because the logic consequence of that would be wrong. In the same way "If we were able to make it to Friday without a test, then we'll know that the surprise test is Friday" isn't an invalid line of thinking. It's simply showing that it can't be possible for the surprise test to be on Friday.

ajuc 15y ago

I didn't assumed Friday probabilty is > 0, nor any other day. I didn't assumed X is not random variable with other options. I just assumed all the other options for sure have probability of 0, so can be ommited, and for days 1..5 I didn't assumed anything other than teacher guaranted. But we are given guarantee that no day will have probability 1.0, if I know it wasn't any day before. From the guarantee we know it won't be Friday. From this we know it won't be Thursday, and so on. From the fact that teacher told us that there will be an exam this week, we know any other options we could add (like "not given at all", or "next monday") have probability of 0. From this all we see that teacher claims were contradictory.

its_so_on 15y ago
Right, you are given the guarantee: "that no day will have probability 1.0, if I know it wasn't any day before". The key is that "if I know it wasn't any day before" CAN'T be a false premise. If the teacher secretly fixes the day at Monday, then the first line of reasoning becomes FALSE. The student is WRONG to argue "If I make it to Friday..." because that's not possible any more than "If I make it to Saturday..." or "If I make it to next week..." All of these are false premises. Arguments from a false premise don't lead to any conclusions. "If your mother is named John, then..." is a mistaken argument, if there is a 0.0 chance your mother is named John (for some reason). It doesn't matter what the other half of the sentence is, if that premise has a 0.0 chance of being true, you can't really deduce anything from it (if it has 0.0 chance of being true). The thing that makes this thing hard is you don't KNOW how the teacher chooses the day. If the teacher always picks Monday, but nobody in the class knows it, they don't know that Friday has a 0.0 chance of happening, they begin to reason "If we make it to Friday morning without a test..." and this argument is simply FALSE. It's about something not possible. It is like saying "If 1 = 2..." The student makes a mistake (if the teacher always gives a test Monday) by saying "If we make it to Friday without a test..." just as the student makes a mistake if they finish the sentence "If 1 = 2..." I mean, I can continue making arguments: "If 1 = 2, then 2 = 4". Is that a "true" statement? Not really. You can't really deduce anything about 2 being equal to 4 or not if 1 can't equal 2. You can't really deduce anything about other days of the week, if it's not possible for the teacher to pick Friday. That is why it matters how the teacher chooses the days.
- ajuc 15y ago
  Nitpicking: "If 1 = 2, then 2 = 4" technically this is a true statement. Any implication starting with false premise is true. But you already know that, and you meant that I cannot use the "then" clause of this implication, because it can be true or false. Main subject: My reasoning resulted with contradiction, which I blamed on assumption (1) or (2), but it can be that there was hidden assumption (3) "there is a chance that Friday will be choosen". When we assume (1) (2) and (3) we have contradiction, when we assume (1) and (2), but not (3), we can't use (2) to prove there's no possible day to choose for teacher, so that case is the solution. OK - count me persuaded. Thanks for disscussion.
- prof_hobart 15y ago
  >"If we make it to Friday morning without a test..." and this argument is simply FALSE. It's about something not possible. It is like saying "If 1 = 2..." Well, yes it is like saying "if 1=2, then 2=4", but that doesn't mean that this line of thinking is an invalid one. It simply means that by showing the result to be wrong, the test at the start is also wrong. The statement isn't attempting to prove that 2=4 by assuming that 1=2. It's saying that we can show that 1 quite clearly doesn't equal 2 because the logic consequence of that would be wrong. In the same way "If we were able to make it to Friday without a test, then we'll know that the surprise test is Friday" isn't an invalid line of thinking. It's simply showing that it can't be possible for the surprise test to be on Friday.