The Unexpected Hanging Paradox

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In this video we take a look at the unexpected hanging paradox, also known as the surprise exam paradox. Almost 100 papers have been published on the paradox, but there is still no agreement on what the correct solution is. Can you solve the paradox?

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Music:
“Constancy Part Three”
Kevin MacLeod (incompetech.com)
Licensed under Creative Commons: By Attribution 3.0

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26 COMMENTS

  1. Stupid paradox. Nice to see brain power was put towards trying to figure out this complete waste of time.

  2. A Friday hanging would still be a surprise, but the surprise is on Thursday.

    Or, on Thursday they tell him it’s postponed until next week. Then hang him on Friday. Surprise!

  3. WHY does that fool think that he can't be hanged on a Friday??! How can't Friday be a surprise? Any day will be a surprise. I can't understand.

  4. Well, if the prisoner was an anxious person he would never be hanged because he would always be expecting it. In that sense… The fault for the paradox lays on the judge then, I guess.

    I get that the whole hanging thing is just an analogy to portray the real paradox: Some event, during an established time period, will only happen when a subject doesn't expect it, while also knowing the conditions for it to happen. If the subject expects it to happen during the whole time period, it by definition can't happen, or else the premisse is false.

    I guess it also deals with consciousness in some way, because… Even though he expected it in some moments it would just slip his mind – if he were to think about escaping, for example. All in all, the sentence being attached to the prisoner's conscience of it is an issue because they couldn't be sure of his consciousness or expectancy at all. Again, the fault for the paradox, as told in this story, lays on the judge.

    Anyway, again, I know the whole prisoner thing is an analogy to portray the real paradox, but it's a bad one, imo.

  5. Wow 200 papers written on this subject, some people clearly have way to much time on thier hands,
    Obviously the prisoners logic is flawed as he is making assumptions based on future outcomes , for instance the prisoner can only say that it can’t be Friday if he is still alive on Thursday , his ‘assuming’ that he will still be alive is floored (cos he might be hanged by then), and therefore no paradox

  6. Easy, when the prisoner rules out a day, it is then a surprise if it is to happen on the day he ruled out due to forward theorising.

  7. Alot of people are saying that the prisoner being surprised to be hung on Wednesday after deducing he couldn't be hung at all is not the answer (I totally agree with because he deduced that he wouldn't be hung anyway after he eliminated them is the surprise when his cell door gets knocked on wednesday) because of what ever reason. What's the level of intelligence of the prisoner? If he manages to work it out and eliminate the days it's a surprise….. and if he doesn't understand (afterall if he was that smart he wouldn't have been caught) then his cell door getting knocked on on the wednesday is still a surprise so judge wins both ways if you ask me.

  8. This whole thing makes no sense

    "For no reason at all I will not be hanged on Friday, but because of that I also cannot be hanged on Thursday, somehow …" and keeps going

    Tell me, does that make any sense to anyone and would anyone think about it like that?

    The prisoner knows for a fact that he will be hanged on a weekday the following week, but he for some reason thinks something else and because of that belief he is surprised because something he knew for a fact was going to happen happened?

    This whole thing makes no sense

  9. if he deduces that a friday hanging isn't surprising, then if he was hung on friday it would be a surprise

  10. Since he concluded that he wouldn't be executed, getting executed would obviously be a surprise 🙄🙄🙄

  11. i am more surprised about the fact that 100 papers were published on this bullshit. and btw i dont see paradox, dunno how the first assumption is ever made, how he excludes friday in the first place. I mean sure i also excluded friday but not because the reason of the video. The reason is that he will be hanged the next week (and today is as i understood Thursday, so it cannot happen the next day, but the next wekk. Based on that, every assumption based on the first assumption is flawed this all video is bullshit.

  12. Seems pretty simple.. Friday can be eliminated due to its finality contradicting the idea of “surprise” but only if all other days occur without incident. Thus each days probability quotient is contingent on the previous day. 1/5, 2/5, 3/5, 4/5 and finally Friday which registers a 5/5 precluding it, but only sequentially. Perceived paradoxes are results of applying a bad theory, mathematical understanding of probability is simply marginally inaccurate. Don’t be sheep

  13. Prisoner knows it can't be Friday IF he makes it to Thursday evening. If. Friday has only been ruled out IF he makes it to Thursday evening. Not before. The earlier in the week it happens, the lower his chances are to guess the day of the hanging. The later along it goes, the higher the chances. Which only means we all know he won't make it till Thursday evening, because that is the only time he knows he can expect the hanging with 100% certainty.
    He has to expect it to a certain extent, given that he knows there are only 5 weekdays left until his hanging. So, if the prisoner makes it till Thursday morning, he could still be killed on Thursday or Friday. He doesn't know which.

    On Sunday evening (before the week of execution), the prisoner could have ascribed a 20% probability for each day. Then, as each day passes, the probability gets distributed among the remaining days. That's why it's no surprise that upon Thursday evening he will "expect" a Friday hanging: There is only one noon left, so it's 100%. On Wednesday evening, it has to be Thursday or Friday: 50/50. So, he can half-way expect either day. Consequently, if the prisoner makes it till Tuesday, he could still be killed on Wednesday, Thursday, or Friday, at 33% each. And Wednesday it was.

    To me it seems as though the word "expect" gets to weasel its way through the thought experiment instead of a more accurate wording. Expectation seems binary (and I would say even expecting something 20% is sort of expecting it), when what we're talking about is "probability of guessing right as the options get eliminated". Almost like the Monty Hall problem.

    Buuuuuut I'm sure I'm the one messing up here. I'm probably just surfing along on a wave of ignorance and imprecise language and thinking.

  14. The fault with his logic is that he conflated the set of possible weekdays with the set of possible hanging days. We'll call the former W, and the latter H. We know from the start that W {Monday, Tuesday, Wednesday, Thursday, Friday}. Using the prisoner's logic, we can know that Friday cannot belong to the H set, as there is the additional condition of the actual day of hanging being a surprise until it is announced to the prisoner.

    However, while Friday is not the day the hanging will take place, it still belongs to the W set, and the drawing pool from which hanging day will be selected is still within the W set. In this way, the H set is shown to actually be a subset of the W set. W { H { Monday, Tuesday, Wednesday, Thursday}, Friday}.

    By removing Friday entirely from his reasoning and reusing the same logic he is actually reducing the number of weekdays, and treating Thursday as the last possible weekday. By changing the sets in this way and applying the logic again he gets this: W { H { Monday, Tuesday, Wednesday}, Thursday} but, as we already established, W includes five days and not four.

    The reason he fell into this error is that he incorrectly framed the argument as to why Friday couldn't be the hanging day in his head. His reasoning was focused on the days before Friday and, by reaching the fourth day he thought that the fifth must be excludes as x number of days have gone before it, but the actual reason Friday isn't the day is because of the days after Friday, that is to say, no days. Friday, in this particular context, shouldn't be framed as the day that comes after Thursday, but as the day with no possible weekdays immediately following it.

    What do you guys think?

  15. Everyone is trying to see it in the same perspective as everyone else but no one realized that the paradox itself doesn’t exists because it’s an attempt at predicting that future which is unquantifiable due to the many possible outcomes

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