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VideoThe Sleeping Beauty Problem (youtube.com)

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[–]likesleague 159ポイント160ポイント  (143子コメント)

I've said this in a couple other comments, but to put it concisely:

The question is faulty. (Or, if you prefer, "poorly worded," intentionally or otherwise.)

The wording is, "What is the probability that the coin came up heads?"

1/3ers think this is asking, "What is the probability that you are being woken up and questioned as a result of the coin turning up heads?"

1/2ers think this is asking, "What is the probability that, when the coin was tossed, it would come up heads?"

Both are incorrect1 interpretations of the original question because the original question is nonsensical. The thought experiment set up is designed to make people think you're trying to answer the 1/3 question, whereas the wording is so close (came vs come, basically) to the 1/2 question that many will try to answer that question.

TL;DR

1 I felt this was a bit unclear. I don't mean that anyone is wrong for reading the question one way or the other, but rather that a definitive 1/2 or 1/3 answer is not an answer to the original question, because the original question is poorly worded and arguably has no answer.

Edit: Many people have messaged me saying that the 1/3ers question is the one that was intended to be asked (and some other descriptions of the problem have worded it more like the 1/3ers question) and that, while the answer seems to be 1/3, the experiment may instead have been intended to spark discussion on the nature and value of probability. I'm no expert on this (I mainly do philosophy on the human mind, which is tangentially related to our understanding of probability) and there are tons of great ideas on it in the comments below me, so I encourage anyone with that position to check out some of the replies to this comment!

[–]dickwagstaff 34ポイント35ポイント  (32子コメント)

TL;DR The answer is 1/2 but you would only be correct 1/3 of the time.

[–]Supperhero 52ポイント53ポイント  (3子コメント)

Actually, you'd be correct all the time because you're talking about what the probability of the result is, not what the actual result is.

[–]NecroticFury 3ポイント4ポイント  (0子コメント)

not based in a situation where they try to integrate an ulterior motive into the experiment, like in this case. First she asks the odds of the coin, then asks the odds of the girl being right. Those are two separate questions.

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    [–]OutOfStamina 3ポイント4ポイント  (0子コメント)

    would only be correct 1/3 of the time.

    Only because you are being asked repeatedly for one event but not the other. The coin flip was 50/50. It's 50/50 for being in one chain or another.

    As sleeping beauty, it's not my fault that one of the two possible outcomes forced you to ask me a 2 or a million times. That doesn't change the odds of being in that scenario in the first place.

    [–]cramduck 6ポイント7ポイント  (2子コメント)

    Another way to look at this is to instead break the test up across multiple subjects. 100 people sit in 100 isolated rooms. I stand in the hall and flip a coin. If it comes up "tails", I go in and tell a randomly-selected two of them the result. If it comes up "heads", I go in and tell a randomly-selected one of them the result. All subjects are aware of the test methodology.

    After I walk into the room, but before I reveal the result of my coin toss, I ask each subject what the odds are that I flipped "heads".

    The coin toss itself operated at 1/2, but the fact that I am asking them the question introduces bias, because I ask it twice as many times when the coin came up "tails". The odds of the coin I just flipped being heads, based on the subject's knowledge, is actually 1/3.

    Does that summarize it pretty cleanly? I sort-of felt that adding amnesiacs and multiple days in the original question just muddied the water.

    edit: typo and clarification

    [–]Mixlop2 5ポイント6ポイント  (4子コメント)

    No you'll be right 100% of the time, saying 1/2 isn't saying 'heads' or 'tails' it's stating that the probability is 1/2 which is always correct. Your comment doesn't make sense.

    [–]dickwagstaff 6ポイント7ポイント  (3子コメント)

    My comment makes exactly as much sense as the problem it is attempting to answer.

    [–]Mixlop2 2ポイント3ポイント  (0子コメント)

    Good point, it is a confusing problem. But picture yourself as sleeping beauty waking up with no knowledge of the day and being asked the probability of a coin flip. You would/should say 50/50.

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      [–]The_Yar 2ポイント3ポイント  (16子コメント)

      That is a contradiction.

      [–]xhable 12ポイント13ポイント  (15子コメント)

      I don't see how it is.

      "What is the probability that the coin came up heads" is 1/2.

      If you say heads every time.. you'll only be correct 1/3 of the time.

      As stated above, the question is faulty.

      [–]Tarandon 1ポイント2ポイント  (0子コメント)

      So these are then two different questions.

      What is the probability that the coin came up heads.

      What is the probability that you are awake and today is not Monday

      The question conflates today being Monday with the outcome of the coin toss.

      Since I haven't erased your memory yet, as far as you know, and you're awake, how confident are you that heads was flipped.

      Is a very different question than:

      What is the probability of the coin being heads.

      [–]The_Yar 1ポイント2ポイント  (0子コメント)

      Maybe were agreeing here, but by definition, your two quoted statements are contradictory. If repeated attempts result in 1/3 outcome A, then outcome A has a 1:3 probability. By definition.

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          [–]WilliamKiely[S] 6ポイント7ポイント  (13子コメント)

          1/3ers think this is asking, "What is the probability that you are being woken up and question[ed] as a result of the coin turning up heads?"

          My answer (or at least the answer that seems intuitive to me) to this question is 1/2.

          Re-wording the same question slightly more clearly:

          When you are woken up during the experiment, to what degree ought you believe that the outcome of the coin toss was heads?

          Halfers say 1/2; thirders say 1/3.

          While some halfers are halfers merely because they mistakenly interpret the question as "What is the probability that, when the coin was tossed, it would come up heads?" others (the genuine halfers) are answering the same question that the thirders are answering, but answering it as 1/2.

          [–]likesleague 6ポイント7ポイント  (2子コメント)

          When you are woken up during the experiment, to what degree ought you believe that the outcome of the coin toss was heads?

          I think that's 1/3 to most people.

          Consider repeating the experiment over and over after itself, so that on the final wake up day of a trial (either Monday or Tuesday depending on whether it's heads or tails respectively) you put Beauty back to sleep and flip another coin. Obviously, long-term she will be woken up due to tails 2/3 of the time (because 1/2 of the time the coin will be tails, resulting in two wake ups, and 1/2 of the time the coin will be heads, resulting in one wake up). It happens to be the same for short term, since the only information that we know is that we are being woken up and that there are three scenarios (heads Monday, tails Monday, tails Tuesday).

          Consider if it were a 3-sided coin that was heads/tails/tails, and if it was heads I would wake Beauty up on Monday, if it was tails1 I would wake her up on Monday, and if it was tails2 I would wake her up on Tuesday. In that case, it seems clear that there's a 1/3 chance of being woken up because of heads, since the coin has 3 sides. The difference between this and a heads/tails coin is that the two tails wake ups aren't independent of each other in the context of the whole experiment, since you have to be woken up twice. The probability is exactly the same, however, because of the information that Beauty knows; nothing about the day. For her waking up, tails Monday and tails Tuesday have no discernible difference and behave exactly as if there was a tails1 Monday and tails2 Tuesday, resulting in a 1/3 chance that she is being woken up thanks to a heads toss.

          Edit: /u/Supperhero commented but deleted it. I won't post his comment but I typed up a response so here it is.

          Two of the scenarios happen in sequence. It's incorrect to add probabilities (25% 25% etc.. I actually did that math for someone else as an analogy in a different comment) because of this. If you want, however, you can think of it like this; when one of the tails scenarios occurs (tails Monday, for instance) the other one is 100% guaranteed to also occur (tails Monday -> tails Tuesday, and if tails Tuesday occurs tails Monday must have also already occurred). Since each tails awakening is guaranteed to happen whenever the coin flips tails (50% chance) each tails awakening actually has a kind of 50% chance. That doesn't make sense, of course (150% chance total) but it gets the idea across I hope (100% out of 150% is 2/3 for tails).

          The problem you're having is that we're not awakening Beauty based on coin flips. If we were, you'd be right that it would be 50/50, since with every coin flip we'd make a check on heads or tails, and of course that's 50/50. But instead we're deducing things about the coin flip(s) from awakenings. You're thinking about the problem like this:

          1. Flip a coin and see what you get.

          2. Wake Beauty once for heads, twice for tails.

          3. The probability should be 1/2, since when I wake her up I'm either waking her up because of heads or waking her up because of tails.

          That's not how it works. This is (slightly modified to get point across).

          1. Flip some coins and make a list of all the awakenings, paired with what coin flip triggered that awakening.

          2. Since a heads triggers one awakening, you expect that half the flips will yield one awakening (H).

          3. Since a tails triggers two awakenings, you expect that half the flips will yield two awakenings (T) (T).

          4. Over time your list will look something like this: (H) | (T) (T) | (H) | (T) (T) | (H) | (T) (T) ... Obviously it probably won't alternate perfectly, but that's for math's sake.

          So now you have a list of the proportion of awakenings (which approaches the true probability as the list approaches infinite length) due to each coin flip result. Note that about 2/3 of the awakenings are due to tails and 1/3 due to heads, simply because each 50-50 coin toss that ends up tails triggers twice as many awakenings.

          1. Beauty has no idea where she is in this awakening list. As it happens, in the original experiment, she's in one little block; either an isolated (H) or one of the (T)s in a (T) (T) block. When Beauty awakens, she is not "picking a block" to awaken in (since she doesn't know whether or not she's already been awakened or even if she'll be awakened again later, which is why the amnesia is important). She's picking a single awakening. There are twice as many (T) awakenings as (H) awakenings and thus there's a 1/3 chance that her awakening was caused by a heads and 2/3 that it was caused by a tails.

          Summary: Tails triggers two awakenings that are 100% likely to occur on each 50% tails flip. In a sense they each have 50% chance of occurring and the heads awakening does too so tails total 100%/150% and heads 50%/150%. 1/2ers (on this wording of the question) are thinking about the problem as trials of coin flips (50-50) when it's actually trials of awakenings (1/3 heads, 2/3 tails).

          [–]spfccmt42 2ポイント3ポイント  (8子コメント)

          "ok beauty, what do you think is the probability that the coin came up heads"

          from the video, it is always 1/2, believe it. Either it is monday and you don't have to go back to sleep, or it isn't. It isn't a mistaken interpretation, it is poor wording if it means something else aside from heads vs tails.

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            [–]Jonathan_Livengood 16ポイント17ポイント  (56子コメント)

            But it isn't a nonsensical question. At least, it isn't obviously nonsensical.

            Suppose I flip a fair coin and hide it under a cup. I ask you, "What is the probability that the coin came up heads?" You might be tempted -- perhaps by some frequentist considerations about single-case probabilities -- to say that there is no probability to assign here. But if you are a personalist, you will ask what your (coherent) degrees of belief are that the coin came up heads.

            If you don't rate your clairvoyant powers too highly, I expect you will assign a degree of belief equal to 1/2 that the coin came up heads. You might also assign a degree of belief equal to 1/2 that the coin is now heads and the same value to the proposition that the coin will be observed to be heads after removing the cup. Though you need not do so, if for example, you think I have a trick cup that manipulates the coin after the initial flip.

            So, how would a personalist think about the question posed in the Sleeping Beauty problem? She would say, "When the experimenter asks, 'What is the probability that the coin came up heads?' he is asking for my coherent degree of belief that the coin came up heads."

            I don't say that this makes solving the problem any easier. There are personalist arguments for the halfer and the thirder positions. What I do say is that the problem cannot be dismissed as easily as you suggest here. The question is perfectly sensible from a personalist perspective AND continues to generate puzzlement.

            [–]grass_cutter 64ポイント65ポイント  (45子コメント)

            There are really two different questions being asked. The probability of the coin being heads is 50%. The likelihood the coin was heads given the fact that you were woken up is 33%. End of story.

            It would be akin to the experimenter only visiting you at all, half the time, when the coin comes heads. His very visit, or the setup, is providing new information.

            [–]dasheea 5ポイント6ポイント  (2子コメント)

            How is the experimenter visiting providing new information? Genuinely curious. Before the experiment, you know nothing new, so the coin being heads is 50%. Once you're woken up and the experiment is standing next to your bed, you still have no new information - whether the coin was heads or tails, you would have woken up the same way. You don't know what the coin flip was and you don't know if it's Monday or Tuesday. You're just sitting up in your bed like a derp. But even before the experiment happened (on Sunday, for example), you KNEW this was going to happen once you woke up, that you'd just be sitting there like a derp. There's no new information gained since Sunday.

            I feel like the 1/3 "frequentist" explanation fails here. It feels like when you wake up, there are 3 possible states and you could be in any one of them with equal probability. Kind of... but not exactly because the 2 tail states (Monday and Tuesday) are completely linked together.

            Imagine a huge tank, like an aquarium. I divide it in half from a bird's eye view. Then, I divide one of the halves into half again, so that one section is 1/2 the size of the whole tank, and then there are 2 sections that are each 1/4 the size of the whole tank. I blindfold you and then throw you into the tank (the tank walls are colored and not transparent. Now I tell you to take off your blindfold. You look around and just see 4 walls around you (let's say you don't know the size of the original whole tank). I ask you what's the probability that you're in the 1/2 section (heads) instead of one of the two 1/4 sections (tails). You have no way to tell because you haven't gained any new information since before being thrown into the tank. Before you were thrown in the tank, you would have said 50%. But before you were thrown in, you knew that once you were thrown in and you took the blindfolds off, you would be looking at 4 walls surrounding you. Now that you're in the tank with 4 walls around you, no new information has been gained. The frequentist explanation can succeed here because we clearly see the tank being divided into half (and one section into quarters, and no section into thirds). There's a 1/2 chance you're in one side of the tank and there's a 1/2 chance you're in the other side of the tank.

            [–]CILISI_SMITH 3ポイント4ポイント  (0子コメント)

            Nicely explained. I hear the question in the latter context because it's being directed at Beauty for her to answer.

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              [–]The_Yar 0ポイント1ポイント  (2子コメント)

              The problem is that questions of probability, particularly these sorts, rely on assumptions about the knowledge and point of view of an observer. Then the Sleeping Beauty problem scrambles those assumptions, which essentially destroys the applicability of classic probability, or at least makes it less intuitive.

              There is also the natural philosophical discrepancy between determinism and probability. If I flip a coin, it lands heads, and then I ask, "what is the probability of it having landed heads?" then one might struggle to answer. Is it 100%, knowing what we know, or does it still make sense to describe a 50% chance of it not being what it obviously is?

              [–]grass_cutter 0ポイント1ポイント  (1子コメント)

              You anyway answered your question. Probability describes one's current knowledge of affairs, not necessarily reality.

              When you shuffle a deck of cards it's quite likely you will assemble an order of cards never before existing in the universe. Hence you sign a 1/52 chance of any particular card being on top of the deck. But really, you physically moved each exact card into place. There was no randomness about it.

              Flipping a coin, if you measured enough variables upon its initial launch you could probably assign a guaranteed result. Probability is our knowledge, nothing more.

              [–]likesleague 0ポイント1ポイント  (4子コメント)

              I don't know much about personalism and I've only talked very briefly about degrees of belief, so I may be off in left field here.

              I believe your interpretation of the original question would inherently effect your degree of belief that the coin came up heads.

              If you interpret the question in the context of a general coin flip (i.e. what is your degree of belief that when I flipped the coin it would come up heads?) you'll say 1/2. If you interpret the question in the context of awakenings (i.e. what is your degree of belief that this question is being asked after the coin having turned up heads?) you'll answer 1/3. The former is obvious, and I can mathematically/computationally demonstrate the latter.

              I don't believe this is a problem about how people interpret this specific question, and I don't think it's a problem about how one constructs belief. Of course you can argue those things, but they would probably be better discussed in a context directly constructed for thinking about them, rather than a question like this where there are mathematical gymnastics that can be done to basically avoid doing any philosophy.

              [–]Brian 2ポイント3ポイント  (3子コメント)

              what is your degree of belief that this question is being asked after the coin having turned up heads

              That's not what's being asked though, or how people are interpreting it - the answer to that question is 100%, not 1/3 after all. Rather, the relevant question seems to be the reverse of that:

              • What is your degree of belief that the coin turned up heads after being asked this question?

              And there does seem to be genuine disagreement between the halfer and thirder positions over this question, rather than the halfers interpreting this as "What is the probability that, when the coin was tossed, it would come up heads?"

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                [–]nhammen 2ポイント3ポイント  (3子コメント)

                1/2ers think this is asking, "What is the probability that, when the coin was tossed, it would come up heads?"

                No, I am thinking this is asking, "What is the probability that the coin came up heads given that I was woken up?" And the answer is still 1/2 because of math.

                Let W be the event that she is woken up, and let H be the event that the coin is heads. Is the question asking P(H|W)? If so, you can apply Bayes' theorem, to get

                P(H|W)=P(W|H)P(H)/P(W)

                But P(W|H)=1 and P(W)=1.

                [–]junerd 4ポイント5ポイント  (1子コメント)

                I agree completely. The question is a play on words.

                1/2ers are talking about coin tosses regardless of an observer, and the 1/3ers are talking about observation caused belief.

                To come up with a clarifying example imagine if instead of waking up more than once the tails outcome killed the sleeping beauty.

                Pre-death the probability is 1/2. After having observed (i.e. still alive) the chance of heads is 100%.

                The above trickery wouldn't work if the outcomes were so black and white. As OP said, the unclear wording and also imo adding the strangeness of waking up multiple times throws people off.

                [–]bullettbrain 1ポイント2ポイント  (1子コメント)

                To me the problem sums up the gamblers fallacy very well. The coin will only ever have 50/50 chance of heads out tails. The actual outcome of the experiment as sleeping Beauty experiences it may be a 1/3 chance of waking up, but ultimately, every time the coin is flipped, there's a 1/2 chance it's heads, and flipping a tails first doesn't change that.

                [–]drukath 1ポイント2ポイント  (0子コメント)

                The opposite is in fact true. You are right that every time a coin is flipped there is a 1/2 chance of it coming up heads, but this is only for future results. Once a coin has been tossed the result is no longer 1/2, but 100% whatever came up. To assume that a past result is still undetermined, or part of the 1/2 of all future tosses is exactly what the Gambler's fallacy is teaching us.

                Anyone thinking the result could be 1/2 has not realised that past results are determined and no longer probabalistic.

                [–]sheeplycow 1ポイント2ポイント  (0子コメント)

                I wouldn't say nonsensical because the whole point of is there are 2 ways you can approach the question thus giving 2 possible answers. I disagree with your conclusion, the whole point about the question is to spark the debate of why can't we solve this question. It isn't supposed to be clear, I would say it's more of a thought provoking question.

                I'd say both of your kind of sub comments are just extensions of the same question because you can interpret the question to mean both, not that the question itself is wrong.

                edit: I preferred the discussion over at /r/math about a month ago

                [–]flatitude2000 1ポイント2ポイント  (0子コメント)

                As a math person (not a philosopher) I find it more than reasonable to assume that a legitimate question is being asked. Therefore your description of the 1/3ers interpretation is the correct interpretation, because asking what you say the 1/2er interpretation would be silly. Everyone knows that a fair coin has a 1/2 probability of coming up heads, and there must be a point to the elaborate set up. And finally because it's an interesting question if assumed that it's

                "What is the probability that you are being woken up and questioned as a result of the coin turning up heads?"

                [–]jabarr 1ポイント2ポイント  (0子コメント)

                I think a good question to ask would be, if over the course of 10 trials, how often would saying "tails" be the correct answer? How often would saying heads be the correct answer?

                Over 10 trials, a perfect outcome would be 5 for heads, and 10 for tails. If someone answered "tails" every single time, they would find themselves correct twice as often as they would by saying "heads". This is because for every event that "tails" is correct, there is a 100% chance that "tails" will be correct for the next event. Meanwhile, if "heads" is answered after every correct "tails" answer, then there is a 100% that they will be wrong. If two experiments are formed, each one having 10 trials, and in each one the person either answering "heads" for every answer, or "tails" for every answer, and the trial only progressing for as long as right answers are called (meaning that if they answer "heads" when "tails" is correct, then they will not be put asleep again to awake on tuesday, instead a new trial will begin with the "initial" coin toss), the person answering tails for every trial will find themselves in an experiment lasting twice as long as for the person who is answering heads for every experiment.

                [–]AnnTauz 1ポイント2ポイント  (2子コメント)

                It has become a weird kind of paradox where both answers are good?

                [–]likesleague 3ポイント4ポイント  (0子コメント)

                In a sense. There are still right ways of applying probability and wrong ways of applying probability, but I honestly feel like the particular wording on this question was a sort of bait to get people to fight over it without realizing it was simply a matter of interpretation.

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                  [–]itonlygetsworse 1ポイント2ポイント  (0子コメント)

                  https://en.wikipedia.org/wiki/Boy_or_Girl_paradox

                  I think is a better example for probability theory since it also addresses how the question is phrased to determine how you would approach the issue.

                  [–]GlutenFreeVodka 1ポイント2ポイント  (0子コメント)

                  The wording is, "What is the probability that the coin came up heads?"

                  1/3ers think this is asking, "What is the probability that you are being woken up and questioned as a result of the coin turning up heads?"

                  1/2ers think this is asking, "What is the probability that, when the coin was tossed, it would come up heads?"

                  The question is not faulty. You're trying to create a word problem where there is none. The answer is 1/3 because every time the coin flips to tails, they ask her twice. It's pretty simple. The probability of the coin flipping to heads (this is not the question asked) is always 1/2 but the probability that the coin actually came up heads this time (this is the question) is 1/3 since they count every tails twice.

                  It's not badly worded. Is perfectly clear.

                  [–]nate_rausch 1ポイント2ポイント  (2子コメント)

                  1/3ers think this is asking, "What is the probability that you are being woken up and questioned as a result of the coin turning up heads?"

                  I don't see what "that you are being woken up and questioned as a result of" adds to precision. Nor what "turning" adds to "came up".

                  I see "What is the probability that the coin came up heads?" as a very precise question to ask sleeping beauty. It only makes sense to ask regarding one "wake-up", and adds no precision to add "questioned" to a question. It has a question mark in the end, it clearly is a question, and "came" clearly relates to one specific event in the past relating to a specific wake-up. The answer is 1/3 since tails gives two wake-ups.

                  In other words you may be correct in analyzing why 1/2ers misinterpret the question. Adding "when the coint was tossed" clearly makes it a question about the coin and not the instances sleeping beauty is waking up - and therefore a a less interesting and different question than the one being asked.

                  [–]likesleague 1ポイント2ポイント  (1子コメント)

                  One of the issues I had with the original wording was the word "came." Past tense is confusing in that question.

                  "What is the probability that [a] coin comes up heads?" is very clearly 1/2 and so trivial that the question isn't really worth asking.

                  "What is the probability that [you were woken up because] the coin came up heads?" is also a very clear question, and clearly 1/3. The reason "you were woken up because" is important is that the event that's happening is Beauty waking up. Asking about the probability that "the coin came up heads" in a context that isn't, by literal wording of the question, related to her awakening, is either nonsense (as I believe it technically is) or unrelated to the event (which yields the 1/2er interpretation).

                  [–]nate_rausch 1ポイント2ポイント  (0子コメント)

                  Yes, I see that you are right now. That is an ambiguity.

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                    [–]Tom_Bedlam 30ポイント31ポイント  (46子コメント)

                    A lot of this problem is confusing probability of an event with probability of successfully completing a test. If sleeping beauty is to pursue a strategy of answering correctly, she should answer tails as two thirds of the time that is the right answer. But that is simply because if the outcome is tails, they ask twice. The probability of the original event i.e. the toss, is still one half. The test and the strategy for the test are separate from that. The number of times they ask influences your score, not physics.

                    [–]Brian 9ポイント10ポイント  (3子コメント)

                    Yes, but the number of tmes they ask seems to influence the likelihood you are in that situation, given the information you have available to you.

                    Eg. let's take a simpler scenario. Every year, the king of a small kingdom performs the following ritual: he secretly tosses a fair coin. If it is heads, he writes a summons for a randomly chosen duke, and, if that duke turns up, rewards him with 150 gold coins. On the other hand, if the coin is tails, he randomly summons two dukes (with the same formal letter), and any that turn up are fined 100 gold coins. The dukes are free to ignore the summons with no penalty if they want. The dukes also don't have time to communicate etc before replying.

                    You are a duke of the king who has just received such a letter. What is your credence that the king's coin landed heads? Should you answer the summons?

                    I think it's much clearer that in this situation, the answer is not 50%. If it were, attending would be a good idea, since you get a net return of 25 gold on average. But in fact, it's a losing proposition - the king is the one who wins this game.

                    This is an example of us conditioning on who we are out of a class of possibilities. The very fact of being a duke who is receiving a letter makes us more likely to be in the class of "dukes who receive the letter in tails years" than those of heads years. So we conclude it's twice as likely the coin was heads.

                    Sleeping beauty differs in that the conditioned states are the same person, spread in time, but with their memory manipulated so that informationally, the situation is equivalent to the dukes case. Should this make a difference? What exactly about it changes things?

                    The probability of the original event i.e. the toss, is still one half

                    Is it though, given the information we have available to us? That's not the case in the Duke's example, because we've received new information that lets us update our probability. The sleeping beauty argument is that the same is true there - that the information is that of finding ourselves in such a situation alters the credence we should assgn, just as findng ourselves in the position of receiving a letter does for the duke.

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                      [–]CMAuGaming 36ポイント37ポイント  (73子コメント)

                      I am generally quite bad at grasping mathematical concepts here, so can someone please ELI5? Why isn't the probability of the coin being heads 1/2? Why does the number of awakenings in the tails outcome matter?

                      The question is, what is the probability of the fair coin coming up heads.

                      If the question was, what is the probability of Sleeping Beauty being woken up on a day where the coin came up heads, THEN I could understand why it is 1/3.

                      [–]Brian 11ポイント12ポイント  (0子コメント)

                      what is the probability of the fair coin coming up heads.

                      No - it's "What is the probability of the fair coin having come up heads, given that you've been woken up and asked this".

                      Clearly, we can give different answers than 50% if we receive extra information after the fact. Eg. suppose you toss a fair coin, and if it's heads, you roll a fair dice, and if it's tails, roll a trick dice that always comes up 6. You show me what the die rolled. It shows "6" - should I still conclude it's 50:50 that the coin was tails?

                      The Sleeping Beauty paradox boils down to whether the fact that we find ourselves in this scenario provides us with such information. The thirder argument is that it does, because it's a situation we find ourselves (indistinguishably) in twice as often when the coin came up heads, and so, on experiencing it, should consider it twice as likely such a scenario occurred.

                      [–]Vicker3000 24ポイント25ポイント  (7子コメント)

                      what is the probability of the fair coin coming up heads

                      This question right here is the core of all the misconception and bickering. This is not the question she is asked (or at least should be asked, if you're phrasing the "problem" correctly). The question being asked instead should be:

                      What is the probability that the (already performed) coin toss came up heads?

                      Let's say I flip a coin, and then show you the outcome. The outcome was heads. I now ask you, "what is the probability that that particular coin toss was heads?" Your answer would be that you are 100% certain that that particular coin toss came up heads. You're there looking at it. You can see that it's heads.

                      You're doing something similar with Sleeping Beauty, but you're not showing her the coin afterwards. Instead you're giving her information that gives her some information about the result of the coin toss, but not complete information.

                      We're asking about probabilities of an event that has already happened. We're giving Sleeping Beauty limited information about that event that has already happened. It is the information we're giving her that determines the probability. Thus the probability of 1/3.

                      [–]fellow_earthing 8ポイント9ポイント  (5子コメント)

                      Wait, what new information is she gaining upon waking up?

                      I'm not quite following you, because to me it seems like the only information SB has about the result of the toss is what she is able to glean from the rules of the game, which were explained to her prior to the toss. There is no information contained in the "waking up and being asked" phase of the game, unlike in your example where the result is shown.

                      This puts me in the 1/2 camp. Many thirders are saying "put yourself in her shoes," but if I were her, my reasoning would go: "Even though there are 3 possible realities I could be inhabiting right now, both of the 'tails' outcomes must necessarily be crammed into the 1/2 likelihood that the coin came up tails in the first place, so they each carry a 1/4 probability of having happened."

                      [–]Vicker3000 11ポイント12ポイント  (3子コメント)

                      Wait, what new information is she gaining upon waking up?

                      The information is that she is awake. Let's construct a new, simplified version of the Sleeping Beauty problem:

                      If the coin comes up heads, wake her up, talk to her, etc. If the coin comes up tails, don't wake her up at all. There is no Tuesday. Now when you wake her up on Monday, you ask her how she thinks the coin toss went. She can tell you with 100% certainty that the coin was heads, given that you wouldn't be asking her that question if the coin had come up tails.

                      [–]qevlarr 0ポイント1ポイント  (0子コメント)

                      That sounds like you are fitting your argument to reach a fixed conclusion.

                      I can get into the 1/2 result from the "no additional information" argument, though I would argue that the fact that you are waking up is that additional information. What I really do not understand, is by what logic you can distribute that 1/2 of coming up Tails over the events Tails-Monday and Tails-Tuesday. You do account for the double trial in the Tails case, but then you don't use this information in determining the probabilities of the coin flip given you are waking up! Really, I can understand why you would say 1/2 on the coin flip, but not the 1/4, 1/4. That does not make any sense. Can you explain?

                      [–]ThinknBoutStuff 7ポイント8ポイント  (0子コメント)

                      The outcome was heads. I now ask you, "what is the probability that that particular coin toss was heads?" Your answer would be that you are 100% certain that that particular coin toss came up heads.

                      I didn't see the difference until you explained it in this way. We aren't asking about the probability of a fair coin. We are asking about the probability which follows from its being flipped. Upvote for clarifying!

                      [–]AggregateTurtle 17ポイント18ポイント  (0子コメント)

                      that is the semantical disagreement happening here, really. I think you understand just fine.

                      [–]drukath 2ポイント3ポイント  (0子コメント)

                      The probability of a future coin toss occurring is still 1/2. Nobody is denying what that is.

                      The question, however, is "what is the probability that the coin came up heads". Now we are talking about a past event. As that event has occurred the probability that it occurred is 1. It happened, and there was no chance that it could unhappen.

                      So now we are not asking about a future probability but a past one. The event that occurred has a 100% chance of having occurred (and the experimenter, for example, will know the result). However Sleeping Beauty does not know the result, so she has to work it out.

                      Now say you were with the experimenter when the coin was tossed. The experimenter has seen the result but you have not seen it yet. This is going to sound a bit weird but given the situation you have no additional information so you know there is a 50% chance it will be 100% heads, and a 50% chance it will be a 100% tails. That gives you a 50/50. It is important to phrase it like that because the event has already happened so in reality it is not 50/50 it is definitely 100% something.

                      Sleeping Beauty, however, has more information. She knows that there are 2 times that she will be woken up with no prior memory that it is a tails, and 1 time it is a head. So at the moment she wakes up she knows it is 1/3 chance of 100% heads and 2/3 chance of 100% tails (remember the coin toss has happened so it is 100% something).

                      Therefore the correct answer to her is 1/3 * 1, or more simply 1/3.

                      [–]pseudo_meat 4ポイント5ポイント  (0子コメント)

                      I think it's because, if you're sleeping beauty, there are three possibilities that you could be in. 2 of them are a tails flip. So what's the likelihood of the third option? 1/3.

                      I definitely agree with the 1/2 people. But, putting myself in her shoes, I can see myself thinking "well, there are three different universes here. The probability of being in one of them (IE the heads one) is one out of three. That's one third".

                      At least that's my interpretation.

                      [–]Flaisse 6ポイント7ポイント  (38子コメント)

                      The premise is that she has no knowledge of her previous awakenings and the day of the week. This means that there are three scenarios that she could be woken.

                      It is monday, and the toss was heads.

                      It is monday, and the toss was tails.

                      It is tuesday, and the toss was tails.

                      When she wakes up, there are two possible scenarios for the awakening that stem from the tails flip, and one that stems from the heads.

                      So the probability of it being heads is only 1/3.

                      [–]johncheswick 8ポイント9ポイント  (15子コメント)

                      Why are we assuming an equal likelihood for all 3 states?

                      [–]Brian 10ポイント11ポイント  (0子コメント)

                      If the coinflip is fair, then all 3 states happen equally often. Eg suppose you were to perform this 100 times, getting 50 heads and 50 tails. You'd then find 50 incidences of all 3 of the states occurring.

                      [–]Blackdragonproject 2ポイント3ポイント  (13子コメント)

                      And that they are exclusive, which does not hold. The sum of all states must be 1 and state 2 occurs if and only if state 3 occurs. So there are only 2 states.

                      [–]Brian 4ポイント5ポイント  (12子コメント)

                      Those states are clearly exclusive No two states can be true at the same time (ie. it can't be monday and tuesday). Did you mean independent here? If so, that doesn't seem relevant given that we're being asked about our perspective from an informationally indistinguishable state.

                      The sum of all states must be 1

                      That seems to argue for the thirder position if all 3 are equally likely (and they will certainly occur with equal frequency in repeated experiments), since the only possible sum of 3 states that have equal probability and sum to 1 is 1/3 for each.

                      [–]It_stands_to_reason 5ポイント6ポイント  (11子コメント)

                      But the second scenario doesn't exist because, at least in this presentation of the problem, she is not asked on the Monday-Tails how the coin landed, only on the Monday-Heads or Tuesday-Tails.

                      edit. I'm leaving this comment the same, but wanted to clarify that she is asked every day, not just at the end. My bad.

                      [–]pajam 12ポイント13ポイント  (10子コメント)

                      Yes she is, she is asked every time she is woken up. Not just on the "last day."

                      The problem is, /u/Flaisse didn't answer /u/CMAuGaming's question. /u/CMAuGaming already recognizes that the probability of the sleeper being woken/asked in a scenario where the coin landed heads is truly only 1/3. But the question of probabilty of the coin landing heads is still simply 1/2.

                      If the question to the sleeper was "What do you think the coin landed on?" then they would guess tails if they wanted a higher chance of being correct. But it still doesn't change the probability of the flipping of the coin. It all depends on the question being asked. And in this case, the question being asked should give the same exact answer every time (1/2) no matter if tails was more likely in the majority of scenarios.

                      [–]It_stands_to_reason 3ポイント4ポイント  (0子コメント)

                      Hmm... I must have missed that. I didn't realize she was going to be asked every time she is woken...

                      [–]Urethra_FrankIin 5ポイント6ポイント  (2子コメント)

                      The video says she is only asked when she wakes up at the end of the experiment what the probability is that the coin came up heads. It doesn't say she was asked every time she woke up. Even so, it's a single chain of events that has been completed from one coin toss, so regardless, there are only two outcomes. Either she woke up once or she woke up multiple times. It doesn't matter how many times she woke up in the tails scenario because the experiment is over when she is asked the question.

                      [–]AnnTauz 2ポイント3ポイント  (0子コメント)

                      Then for her the experiment will never be different, she will always think of the probability of 1/2. The only difference is for the observer who knows the answer, and for her it will simply not matter until she is given with the real answer, if the experiment is made a million times I think it won't change, it has become a matter of perspectives and how badly the tester makes this test affect their needs for knowing the answer.... My brain hurts

                      [–]CILISI_SMITH 1ポイント2ポイント  (0子コメント)

                      The video says she is only asked when she wakes up at the end of the experiment what the probability is that the coin came up heads. It doesn't say she was asked every time she woke up.

                      Actually it doesn't clarify either. We're just told she's asked "when she wakes up" which could mean every time or only the final time. All arguments here are over ambiguity in the language used.

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                          [–]Fox_and_Otter 1ポイント2ポイント  (5子コメント)

                          its a silly notion at best. The probability of the coin being heads or tails is always 1/2. Unless i seriously misunderstood this, then the probability of the coin being heads or tails is 1/3rd because there are 3 possible outcomes from the observers perspective. The silliness here is assuming that because there is an unknowing observer, then the probability changes from the perspective of the observer. Personally I think this is wrong, the probability of heads or tails is the same. I also think its wrong from the observers perspective. I think that there is still a 1/2 chance of it being heads or tails, but from the point the question is asked the probability of it being monday is 3/4's and the probability of it being Tuesday is 1/4. If i've missed anything i hope someone lets me know. Who knows, this kind of thing might lead somewhere one day, I wonder what the probability of that is.

                          [–]itisike 1ポイント2ポイント  (3子コメント)

                          its a silly notion at best. The probability of the coin being heads or tails is always 1/2. Unless i seriously misunderstood this, then the probability of the coin being heads or tails is 1/3rd because there are 3 possible outcomes from the observers perspective. The silliness here is assuming that because there is an unknowing observer, then the probability changes from the perspective of the observer. Personally I think this is wrong, the probability of heads or tails is the same. I also think its wrong from the observers perspective. I think that there is still a 1/2 chance of it being heads or tails, but from the point the question is asked the probability of it being monday is 3/4's and the probability of it being Tuesday is 1/4. If i've missed anything i hope someone lets me know. Who knows, this kind of thing might lead somewhere one day, I wonder what the probability of that is.

                          So if she is told that it's Monday, what is her posterior on heads after a Bayesian update? How is that case different than when she's only woken on Monday regardless of coin flip?

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                            [–]PM_ME_KIND_THOUGHTS 1ポイント2ポイント  (14子コメント)

                            imagine yourself in her shoes.

                            you wake up, not knowing even what day it is.

                            If instead of telling you to name the odds that the coin came up heads or tails, the people made you guess which one DID come up, how would you go about guessing?

                            you could be stubborn and say "technically it's still a 50/50 chance it was heads, so it doesn't matter what I guess," but, really, to intelligently determine whether you should guess heads or tails, you need to figure out not what the probability is that something COULD HAVE happened, but what the probability is that one of the two events DID happen in the reality that you find yourself a part of. when you think of it like that, you would probably come up with the fact that there is really a 2/3 chance that it came up tails.

                            [–]hairam 5ポイント6ポイント  (13子コメント)

                            But the issue I see is, regardless of how many chances she has to guess if it was heads or tails, the initial problem only has two cases. She wakes up Monday, or, she wakes up Monday, goes back to sleep, then wakes up Tuesday.

                            If upon getting tails there were a second coin flip, I can see this being a more complicated issue, but I just don't understand how we can be philosophical about math in this either/or situation. This isn't the Monty hall problem, it's a simple case of two options...

                            I guess I can perhaps see some philosophy behind what beauty believes to have happened, but that doesn't change the 50/50 chance of her either being woken up once, or woken up twice...

                            [–]PM_ME_KIND_THOUGHTS 3ポイント4ポイント  (6子コメント)

                            if you were in sleeping beauty's position and somebody made you guess which way the coin came up, under punishment of death, would you not come to the conclusion that the odds are greater it came up tails?

                            [–]galosheswild 1ポイント2ポイント  (0子コメント)

                            Depends how the rules are structured. If I they are explained to me on Sunday night, and I can choose my strategy and remember it, then I will survive 50% of the time whether I go with heads or tails.

                            [–]dersteppenwolf 1ポイント2ポイント  (3子コメント)

                            Under the penalty of death it wouldn't matter what you guessed because you have a 50/50 chance of guessing right. However, if they penalized you $10 for every wrong answer you should guess tails.

                            [–]TychoCelchuuuΦ 3ポイント4ポイント  (1子コメント)

                            Imagine that you are in a room and I am in another room, flipping a coin over and over. Every time the coin comes up heads, I text you a picture of the coin. Every time the coin comes up tails, I make a check-mark on a piece of paper, and whenever I've made ten check marks, I text you a picture of the coin. So in effect for every 10 text messages with a picture of the coin having come up heads, you also get 1 text message with a picture of the coin having come up tails. Of course, the messages aren't evenly spaced: sometimes you get 9 heads messages then a tails message, sometimes you get 6 heads then a tails, etc.

                            Now imagine someone else comes into the room and pulls a gun on you. They say "I want you to predict whether the next message is going to have a picture of the coin showing heads or the coin showing tails. If you get it wrong I shoot you."

                            One thing you could say is that "well every time Tycho flips the coin, it's 50/50, so whatever message I get, that coin is a 50/50 heads/tails coin, so I'll just make a random guess."

                            But that would be stupid. Surely you ought to guess heads. It's more likely that you'll get a message about a heads coin than a tails coin.

                            This illustrates the point at issue here: if you only know that a coin is being flipped, then you would just make a 50/50 guess. But if the results of a coin flip determine the information you get about the flip, then sometimes this can change what you will guess.

                            Obviously in the sleeping beauty case the rules are more convoluted than "tell you about all the heads but only 1 in 10 of the tails." But we could recreate the sleeping beauty rules with text messages. If we did so, I think you would see why it would make sense to guess (under penalty of death) something other than 50/50.

                            [–]moolah_dollar_cash 0ポイント1ポイント  (0子コメント)

                            If we had a lot of sleeping beauties to kill off and a gun and did this experiment and a third said heads constantly, a third woke up... Flipped a coin and said what that coin said and a third said tails constantly then...

                            Approx half the heads peeps would die.

                            Approx five eighths of the coin flipping beauties would perish

                            Approx half the always said tails gals would not make it through this particularly cruel thought experiment.

                            So it depends if you''ve decided to be flippant before or after you've been woken up. Flippant before and you're right flippant waking up and you're wrong.

                            So.. if we imagine we have been abducted and didn't have this thought experiment explained to us before we are woken up to only then have it explained then the probabilities are different.

                            If we presume (which we can't really but oh well) that if we have been woken up once before we would've said heads or tails on a coin toss then there is a 3/4 chance that we are being woken up for the first time and a 1/4 chance we are being woken up for a second time. That means that if we say heads now there is a 3/8 chance we are right if we say heads and a 5/8 chance of being right if we say tails.

                            Now if we're sensible and think we would've figured this out if we had been woken up before then we can say the probability that on any previous decision we would've said tails is now very very close to 1 instead of 1/2 then the probability that this is the second waking up 1/3 instead of a quarter making the chance of heads being right 1/3. So I personally would always say tails.

                            [–]hairam 0ポイント1ポイント  (0子コメント)

                            The odds are not greater that it came up tails, but, if I'm being asked each time I wake up whether it's heads or tails, the odds of correctly guessing which it was could perhaps be in my favor should I guess tails?

                            So, okay. I flipped a coin 200 times, got heads 103. Roughly 50% (I mean, surely we aren't arguing the probability of the coin being heads or tails as 1/3). But if for each tails I get 2 opportunities to guess, that gives more opportunity to be correct when guessing tails. I don't know if I'm doing this right, but using my 200 coin flip, that's 97 times that I got tails, but multiplying that by 2 because I get 2 guesses each tails, that's 194 opportunities to guess (plus the 103) is 297 opportunities to guess, out of which, I would be correct guessing tails ~ 2/3 of the time.

                            So yeah, it's semantics? My math isn't rock solid, so please correct me if I did it wrong, but, unless the philosophical issue is "how do you pose the question" I'm still a little confused about where the relevance to philosophy comes in (now, after more clearly understanding how the answer to the question varies depending on semantics).

                            [–]itisike 1ポイント2ポイント  (5子コメント)

                            Probability is subjective. You're right that from anyone else's perspective, that probability is 1/2, but for SB, it's 1/3.

                            (Also, consider her Bayesian update upon hearing that it's Monday. If she should previously think that heads/tails were equally likely, then she'd need to think differently then, but that seems inconsistent with the same naive intuition that's leading you to insist the probabilities must be equal.

                            [–]hairam 0ポイント1ポイント  (4子コメント)

                            Probability is subjective.

                            How. How is probability subjective. If you mean semantic, depending on how you frame it, yeah, absolutely. But I fail to see how math is subjective. I'm legitimately asking you this.

                            the same naive intuition that's leading you to insist the probabilities must be equal

                            Ouch. I'm not the best with probability but still. Whew.

                            Did I miss the stipulation that they're asking her each time she wakes up what she thinks was achieved? Like, if they got tails, they wake her up on monday, ask her which she thinks it is, then wake her up on tuesday and ask her what she thinks it is?

                            As someone else said, it seems like this simply comes down to semantics. There's the probability of h v t, then there's the probability of her answering correctly, if she's being asked each time they wake her up. If being asked each time they wake her up, then 2 of the times it will be monday... but even then, they either got heads or tails, so that only makes sense if they're asking her each time they wake her up if she thinks it's monday or tuesday.

                            This is turning more and more into me just thinking as I type, so I'll stop it there - feel free to build upon what I've said, answer my questions (particularly about the stipulations of the situation - when they're asking her, what precisely they're asking her. I can rewatch the video too/try to find how this was posed in its original form) or ask me to clarify if my thoughts weren't followed to completion here.

                            [–]itisike 1ポイント2ポイント  (2子コメント)

                            /u/Brian did a great job explaining what I meant by "Probability is subjective", so I'll just focus on the rest of your points. (Also see http://lesswrong.com/lw/oj/probability_is_in_the_mind/ and http://lesswrong.com/lw/s6/probability_is_subjectively_objective/, which might help clear up some confusion).

                            Like, if they got tails, they wake her up on monday, ask her which she thinks it is, then wake her up on tuesday and ask her what she thinks it is?

                            Yes, and they wipe her mind in between so she doesn't remember being asked.

                            Let me restate the problem, because you asked:

                            SB is put to sleep. A coin is flipped. If it is heads, SB is woken up on Monday, asked what she thinks the coin was, then put to sleep forever. If it is tails, SB is woken up on Monday, asked what she thinks the coin was, then put to sleep. We then give her some memory pill so that she is reset to the state she was in at the beginning of Monday. On Tuesday, we wake her up again, asked her what she thinks the coin was, then put her to sleep forever.

                            See https://en.wikipedia.org/wiki/Sleeping_Beauty_problem for more.

                            There's the probability of h v t

                            If you want to know the probability from the subjective view of someone not in the experiment, then yes, it's 50/50. We're talking about her subjective probability expectation, which is allowed to be different.

                            [–]Brian 2ポイント3ポイント  (0子コメント)

                            How is probability subjective

                            The issue is that it's contingent on mind-dependant facts: the information we have available. The application of probability is objective, but it remains the case that, say, Alice could assign a 50% likeihood to something and Bob could assign a 10% likeihood to the same scenario, and we can't say either has made an incorrect statement, so long as they've done the maths right.

                            You may say we could say the one with more information is correct, or that they're both wrong, and the answer is 35%, because you know everything Alice and Bob know, and also know some other stuff. However, this is problematic, because you'd also have to conclude that you are wrong too, because someone who knew all the information would assign a probability of either 0% or 100%, because they'd also know the truth of the matter. But saying all probabilities except 0% and 100% are always wrong seems to diminish the usefulness of probability somewhat!

                            We could take a more frequentist tack, and talk exclusively about frequencies of populations instead, and forbid ourselves from ever saying something like "The likelihood this fair die will roll a 6 is 1/6", instead only saying "If you roll a large number of fair dice, 1/6 will show 6" But this seems rather limiting, and doesn't allow for a lot of what we use probability for.

                            Rather, we can talk meaningfully about probabilities if we instead view them as a relationship between what we know/believe and how likely we should consider something to be true, given those beliefs. Alice has indeed assigned the correct probabilitiy from her perspective. So have I, and so has Bob, despite these all being different, because those assignments are ultimately statements about our subjective perspectives.

                            "Subective" is maybe a bit misleading, since this certainly doesn't mean "anything goes". Ie you can still be wrong, even given your beliefs (eg. you miscalculate the odds) - I've seen it referred to as "subjectively objective" before for this reason. But the point is really, that when we're speaking of the probability of a single event, we're ultimately talking about the probability given certain beliefs / information.

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                              [–]WilliamKiely[S] 7ポイント8ポイント  (11子コメント)

                              2:14:

                              A further intuition pump to make 1/3 feel even more like the right answer was proposed by philosopher Nick Bostrom. Basically, it asks you to imagine that in the tails world Sleeping Beauty is not woken up twice, she is woken up a million times in a row. In this scenario, if you imagine being woken up and being asked 'what is the chance that the coin came up heads?' it feels much more absurd to say 1/2 given that you know that out of all the awakenings only one out of a million and one are in the heads world.

                              For me this modified problem acted as an intuition pump for the halfer answer rather than the thirder answer. The thirder view would be that the probability that the coin came up heads is only 1/1,000,001. Yet I can't imagine myself being that surprised if it actually came out heads.

                              A further modification to make an even stronger intuition pump for the halfer position is as follows:

                              /u/WilliamKiely's Variant of the Sleeping Beauty Problem

                              Instead of a fair coin, imagine that a fair million-sided die is tossed. If it lands on 1-999,999 then Sleeping Beauty is only woken up on Monday (as in the heads scenario in the original problem). If it lands on 1,000,000 then Beauty is woken up a trillion (that is, a million million) times.

                              For this version of the problem:

                              The halfer position is that Beauty should believe that the probability that the million-sided die came up 1-999,999 is 999,999/1,000,000.

                              The thirder position is that Beauty should believe that the probability that the million-sided die came up 1-999,999 is only ~1/1,000,000 (Note: the exact probability would be 999,999/1,000,000,999,999).

                              Imagining myself in Beauty's position in this modified problem results in an even stronger intuition that the halfer answer is correct. I would expect it to come up 1-999,999 and would be quite surprised if I looked at the die and it said 1,000,000. (If it did say 1,000,000 I would be quite surprised one trillion awakenings in a row, but I don't think the fact that I'd be surprised a trillion times means that my credence that the die was not on 1,000,000 was too strong. Rather, I think that is just what happens when you keep erasing my memory.)

                              The thirders on the other hand must maintain that they would be quite surprised to look at the die and see any number below 1,000,000. I don't understand this intuition. All that would have had to happen was for the dice to be tossed the first night of the experiment and not land on 1,000,000.

                              EDIT: Typos.

                              [–]itisike 3ポイント4ポイント  (4子コメント)

                              Intuition breaks down with such large numbers (see: Pascal's Mugging). Your brain can't represent 1000000 and rounds it down (see: scope insensitivity).

                              So while I share that particular intuition (and the related one in the Presumptuous Philosopher problem), I think it's a feature of those large numbers messing up our intuitions. (There's also the issue of uncertainty; if you're not completely sure of all aspects of the scenario, that can easily change the answer with these sized numbers, and while the problem is stated with surety, the brain "knows" that things are uncertain. To be explicit, your intuition refuses to assign a super-low probability to something that started out with a probability of 1/2, possibly in part because of uncertainty over whether any mistakes were made. If the probability of a mistake being made (or of SIA being wrong) is even 1/1000, that still makes for a far higher probability than your 1 in a trillion. So it's possible that you're doing a similar calculation subconsciously, and therefore rejecting something because of intuition despite the intuition not really applying.)

                              [–]WilliamKiely[S] 1ポイント2ポイント  (2子コメント)

                              Yes, great post. I am now a thirder despite still intuitively feeling that the coin is highly likely to have landed on heads when imagining myself waking up in Beauty's position in my version of the problem. The numbers are so large that I would probably be incredibly old upon waking up making it more difficult to imagine and make my intuitions less accurate. If I magically found myself in the position I think I could now persuade myself to expect it to come up tails, despite the fact that my initial intuitive react was that it was highly likely to have come up heads, due to the fact that I now feel I have a very good understanding of the logic behind the thirder answer and believe it is correct.

                              [–]itisike 1ポイント2ポイント  (1子コメント)

                              The numbers are so large that I would probably be incredibly old upon waking up making it more difficult to imagine and make my intuitions less accurate.

                              To fix that particular barrier to intuition, you can use the incubator version of Sleeping Beauty. In a nutshell, instead of being woken multiple times, you are cloned and put in separate rooms.

                              [–]drukath 1ポイント2ポイント  (0子コメント)

                              Great post.

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                                [–]StevenMaurer 4ポイント5ポイント  (5子コメント)

                                If Sleeping Beauty is asked "What is the possibility of the coin coming up heads?", the correct answer is 50/50.

                                If instead, Sleeping Beauty is told "Let's wager a dollar on whether the coin is heads or tails, which do you want?", the correct answer is "tails". Not because tails comes up more often than heads, but because when it does, she wins two dollars, once for each day she is woken up.

                                [–]drukath 2ポイント3ポイント  (0子コメント)

                                The answer is 1/3rd for Sleeping Beauty, and this is because future probability and past probability is not the same. The question in the video is "What do you think is the probability that the coin came up heads?". This is a past tense question for a specific event that has occurred.

                                Let's try something similar. Let's say that I told you I was going to toss a fair coin, and if it came up heads I'd punch you in the face, and if it came up tails I'd give you $100. I toss the coin in secret and then give you $100. What is the probability that the coin came up heads? Well forgoing that I am a dirty liar you'd say that the coin came up tails, so it was 0% heads.

                                Note that I am not asking you what the next coin toss will be, but what a past coin toss was. New information is introduced because the result of the coin toss determines the next action that can reveal it.

                                Another more obvious one is also possible. Let's say I tossed a coin in front of you and it came up heads. Now I ask "What is the result of that coin toss?" You'd not say it was 50/50 because the event has occurred and you can see it.

                                In the Sleeping Beauty Problem the chance that the coin will come up heads before it is thrown is indeed 1/2. But the question to Sleeping Beauty is not that, it is what was the probability that it came up heads. Given that there is only 1 situation for heads that she would have woken up with no intervening memory, and 2 situations for tails then the answer is 1/3rd. Because we're not really asking what the probability of the coin toss is, but the probability of which of the 3 situations you have been woken up in.

                                [–]Mixlop2 2ポイント3ポイント  (0子コメント)

                                The answer is 1/2 in every scenario.

                                [–]TheCountMC 2ポイント3ポイント  (1子コメント)

                                It always amazes when this "problem" is brought up, how much discussion and disagreement results. Probability is mathematically defined and well understood, but no one ever bothers trying to define a probability space with outcomes, events and corresponding probability measure.

                                This has been discussed on reddit before:

                                https://www.reddit.com/r/math/comments/3915xh/the_sleeping_beauty_problem/

                                https://www.reddit.com/r/math/comments/39bl2g/about_the_sleeping_beauty_problem/

                                Even in the r/math, few people bring up the mathematical definition of probability. Everyone wants to rely on their intuition about probability in a situation their intuition is not familiar with.

                                Shameless self-post-promotion

                                tl;dr - There are two ways to define a probability space consistent with the experiments. One of them (the one with three outcomes) has the caveat that single experiments are sometimes not possible which makes the two outcome 50/50 space better, IMHO.

                                [–]SirDucky 2ポイント3ポイント  (6子コメント)

                                Okay. I'm a mathematician who studies probability and statistics so here are my two cents. We never make it explicit when sleeping beauty will be questioned. However, assuming that she is questioned every time she is woken up, we have a classic sampling bias. The correct answer would be 1/3. This can be shown in both a frequentist or bayesian approach, but in this case a frequentist approach will serve since the approach is repeatable.

                                If the coin comes up tails, once, we know the next time she is questioned it will also be tails. Thus if we were to repeat this experiment indefinitely and construct a series of observations by sleeping beauty, all of the tails would be "repeated" because they are dependent on the previous value. So it might look something like this:

                                HTTHHTTHTTTTHTTHHHTT

                                That's because the coin isn't being flipped again if sleeping beauty wakes up to tails and then goes back to sleep.

                                If we were to repeat this process indefinitely, it would also be a good example of a hidden markov chain. Imagine that there are 3 states:

                                • H1 to signify heads on the first waking
                                • T1 to signify tails on the first waking
                                • T2 to signify tails on the second waking

                                This begets the simple transition and emission matrices:

                                   H1  T1  T2     H T 
H1 0.5 0.5 0   H1 1 0
T1 0   0   1   T1 0 1
T2 0.5 0.5 0   T2 0 1

                                From there a traditional analysis will yield the 1/3 answer.

                                The fallacy with the 1/2 answer is that although the coin is indeed unbiased, sampling tails is twice as likely. Thus , even though the flip is unbiased, the observation is. So while it's true that you're not waking up with any more information, the experimental setup should be all you need to begin with. I didn't know this was a "bitter" debate among anyone, but anyone who is getting bitter about it should go back and study probability some more. It's a non-problem.

                                [–]HeyKidsFreeCandy 2ポイント3ポイント  (1子コメント)

                                Im honestly a little disappointed though at how moderated the r/philosophy content is becoming. I very recently tried to post another excellent video of hers only to have it immediately removed for not being philosophical content. Honestly, if the community is just going to share trivial ethical and moral queries without any real discussion of how to rationally apply them to daily life, then what is the point? I'm not saying we should turn into r/philosophyofscience, but I don't think the mods should just immediately take down posts which give brief, concise introductions to such foundational ideas of philosophy. I always wondered why I never see content on here outside of Theory of consciousness articles, which, while fascinating, is one of the least essential philosophical questions, and moral conundrums which are more interesting puzzles to chew on, rather than philosophically argumentative content. I hope this sub can pick back up.

                                ALSO: It's great seeing Julia Galef on here! If people want to check out more of this rationality-oriented content you should check out LessWrong.com, to which she is a contributor!

                                [–]WilliamKiely[S] 2ポイント3ポイント  (3子コメント)

                                In his book Anthropic Bias, Nick Bostrom writes:

                                Synthesis of the 1/2- and the 1/3-views

                                The account presented here shows how we can accommodate both of the rivaling intuitions about what Beauty’s credence should be when she wakes up.

                                On the one hand, the intuition that her credence of Heads should be 1/3 because that would match the long-run frequency of heads among her awakenings is vindicated if we assume that there is an actual series of experiments resulting in an actual long-run frequency. For there are then many observer-moments that are outside the particular run of the experiment that ß2 is in whilst nonetheless being in ß2’s reference class. This leads, as we saw, to Pß2, 4, 6(Heads|e2) ˜ 1/3.

                                On the other hand, the intuition that Beauty’s credence of Heads should be [1/2] is justified in cases where there is only one run of the experiment and there are no other observer-moments in the awakened Beauty’s reference class [other] than her other possible awakenings in that experiment. For in that case, the awakened Beauty does not get any relevant information from finding that she has been awakened, and she therefore retains the prior credence of 1/2.

                                This appears wrong to me. What if there is an "actual series of experiments" that is only one experiment long? Would Bostrom then say that the probability is 1/3 (as the second quoted paragraph would suggest) or 1/2 (as the third quoted paragraph would suggest)?

                                I don't understand why Bostrom thinks the answer is 1/2 or 1/3 depending on whether the experiment is performed once or many times in series. That just seems wrong to me. Am I misunderstanding Bostrom's position?

                                [–]microli 0ポイント1ポイント  (0子コメント)

                                What matters is the rules for how many times the experimenter will ask the question. Suppose that sleeping beauty decides to always answer tails. If the experimenter asks the question everytime he wakes her up, then she gets the question right 2/3 of the time because there are 2 trials everytime there is a tails and only one trial everytime there's a heads. But if the experimenter only asks the question once per coin flip, then the extra trial for tails goes away and we're back to 1/2 probability.

                                What's even more interesting is that the probability can also be between these numbers if it's not set beforehand whether one or the other method is used for the experiment. If sleeping beauty is not told how the experiment is run, she would have to assume equal probability for both methods, in which case the probability for tails being right is the average of the two above probabilities (1/2 + 2/3)/2 = 7/12.

                                [–]drukath 0ポイント1ポイント  (0子コメント)

                                You're not misunderstanding his position, and yes he is wrong. He is mixing up the future probability of a coin toss (1/2) with an event that actually occurred (1), and our ability to deduce what that event was (1/3).

                                コメントをもっと表示する (1 返事)

                                  [–]ifthisdoesntfitillki 2ポイント3ポイント  (0子コメント)

                                  Perform experiment 10 times: 5 times she awakens on a heads Monday. 5 times she awakens on a tails Monday. If she awakens on a tails Monday she will definitely awaken again on a tails Tuesday. On 10 out of these 15 days, a tails flip occurred, therefore she must answer p(Heads)= 2/3 to be correct.

                                  [–]spfccmt42 1ポイント2ポイント  (3子コメント)

                                  I think there is a wording problem, the probability that the coin came up heads is obviously 1/2. So simply asking her that is simply 1/2. If you were to say, what is the probability that it is monday AND you are not going back to sleep, that is 1/3.

                                  [–]drukath 0ポイント1ポイント  (2子コメント)

                                  The probability that the coin came up heads is 100% what actually happened. You cannot apply future probabilities to past events. Past events happened with 100% probability, because they actually happened.

                                  The question is trying to ask you to deduce what this 100% was based on the information that you have. Once the coin is tossed it is fixed. What is 1/2 before the coin toss is 1/3 after the coin toss to SB given the information she had. For example to the experimenter the result is neither 1/2 or 1/3, but 100% what actually happened.

                                  [–]spfccmt42 0ポイント1ポイント  (1子コメント)

                                  that is even worse wording, and wrong. We are asking the subject what the coin toss probabilities were, and she doesn't know if it is monday or tuesday or if she is going back to sleep again, so she doesn't know what the coin toss was. The question wasn't put "to the experimenter".

                                  [–]chcampb 1ポイント2ポイント  (9子コメント)

                                  So, thinking about it, it's only a problem because they are conflating the events.

                                  The probability of the coin being heads per sequence is 1/2. The probability of the coin being heads per wakeup is 1/3. In this case, the event being measured is the coin flip, which exists only at the beginning of the sequence.

                                  To say that there is a disagreement as to what the "probability of the coin being heads is" is really to say that there is a disagreement as to whether to frame the problem as being per sequence or per sub-event within the sequence.

                                  Obviously, I could give any ratio of sub-events and change the probability to my liking if you allowed someone to choose the "per sub-event" method. So, I am not convinced that this is the correct way to approach the problem.

                                  However, if you said "what is the probability that SB is in one of the states", that would be 1/3 for obvious reasons.

                                  [–]Oznog99 1ポイント2ポイント  (8子コメント)

                                  To extend it, what if she was told "we will NEVER wake you if it comes up heads. If it's tails, we wake you up."

                                  When she's woken up, she knows it's tails. It has to be. If a betting parlor said "we'll pay out 100:1 odds if you bet on heads after you wake up and it turns out to be heads!", well, she wouldn't take it.

                                  In the wording, it's not actually KNOWN. But a betting parlor offering 1:1 odds to her on every wakeup, she would trend to make money if she always bet on tails.

                                  [–]chcampb 0ポイント1ポイント  (7子コメント)

                                  No. There are still two states. You are asking "What is the probability of being in the state of awake after flipping a tails?" and the answer to that is 100% because that's how you defined it.

                                  The question is asking the person being woken up, "what is the probability that tails was flipped?"

                                  Tails has a 50/50 chance of showing up. What happens after that is not the probability of the coin flip, but the state distribution which allows you to infer the result of the 50/50 coin flip. That is all.

                                  [–]Oznog99 0ポイント1ポイント  (6子コメント)

                                  The question is within the context of the observer. That's key. The original probability was, and always will be, 50/50. But in the context of her observation, she should bet her money on Heads only if they offer 3:1 odds payout. Otherwise it's a loss trend. Factually speaking, after playing this game 1000x over, she'd lose everything over time if she bet on Heads with even odds. She should always bet on Tails.

                                  It's because Tails will be observed twice, as a rule. Observer bias. The original flip is 50/50, yes. But factually speaking she would trend heavily towards loss on any "Heads" bet with even odds presented.

                                  [–]chcampb 0ポイント1ポイント  (2子コメント)

                                  Right, I agree. But that doesn't make it correct to say that the original flip is 1/3. If you asked the observer "What was the probability that tails was rolled?" (question A) They have to say 50/50. But if you ask them "What is the probability that you are in a state having resulted from a tails roll?" (question B) then you can respond 1/3 and be correct.

                                  You might say that it's a paradox but it's not, really. People are just asking the wrong questions. Of course if you ask question A and apply the meaning of question B, then of course you will confuse people.

                                  [–]Oznog99 0ポイント1ポイント  (1子コメント)

                                  It would be better to phrase the question as "at what odds should she take a bet for Heads?" It's not 2:1 (bet $1, end up with $2 if you win). She should only take Heads if it's >3:1 (bet $1, end up with $3= $2 profit).

                                  [–]chcampb 0ポイント1ポイント  (0子コメント)

                                  Right but that is just rephrasing question B. Because she needs to understand her states and how they can infer the original flip states.

                                  コメントをもっと表示する (3 返事)

                                    [–]Rebel_de_la_Foret 1ポイント2ポイント  (1子コメント)

                                    Here's how I see it:

                                    She's saying that nothing changes, in the experiment, between the moment Sleeping Beauty falls asleep and the moment she wakes up. So, the probability remain at 1/2 all the time. I think that the fact that she wakes up add to her information about the experiment. Something just changes, you can't continue with your first probability, which was 1/2.

                                    I don't know if I explain my thoughts well, but basically, Sunday the probability to flip heads is 1/2 and now because she woke up, the probability becomes 1/3. Well that's what I tell to myself when I'm trying to sleep and this problem bugs me

                                    [–]xyzz 1ポイント2ポイント  (0子コメント)

                                    but, she knew she would wake up the night before... its not a surprise or new information in my books

                                    [–]klockwork 2ポイント3ポイント  (6子コメント)

                                    I'm going to put you to sleep. Once you are asleep I will flip a coin, if the coin turns up heads, I will wake you up. If the coin turns up tails, I will flip another coin that has two identical sides marked tails and then wake you up. What is the probability that there is a coin turned up heads?

                                    Back to the original Sleeping Beauty there are two independent events - the flipping of the coin, and the waking up once or twice. The odds that you have just been woken up with heads is 1/2. The odds that you have just been woken up the first time with tails is 1/4, the odds that you have just been woken up the 2nd time with tails is 1/4.

                                    Edit - I retract my position. Code proves a random guesser to be correct 33% of the time when guessing tails, and only 17% of the time when guessing heads.

                                    https://jsbin.com/pekanosoyu/1/edit?js,console

                                    [–]soderkis 0ポイント1ポイント  (2子コメント)

                                    I guess one reason the Sleeping Beauty problem causes a kerfuffle is how we go on to explain her reasoning. Before she is put to sleep, she should believe it is 50-50 heads or tails. She knows she is going to be woken up, so for her, when she is woken up she knows nothing new. So how can she motivate that she should believe that it is more probable that it is tails than heads?

                                    I guess one way to try to answer this is to say that she knows she is more likely to be woken up on a Thursday. But this just causes further annoyance, because she hasn't learned anything new when she is woken up.

                                    [–]The_Yar 1ポイント2ポイント  (1子コメント)

                                    She knows that today could be Tuesday, and if today is Tuesday, it was definitely tails. She doesn't know that before or after the experiment.

                                    コメントをもっと表示する (3 返事)

                                      [–]unverified_user 1ポイント2ポイント  (0子コメント)

                                      Imagine that I'm in a room with a coin, and you're in a room with a bell. I will flip the coin in my room, and you won't be able to see it.

                                      If the outcome is heads, there's a 50% chance that the bell will ring. If the outcome is tails, there's a 100% chance the bell will ring.

                                      I flip the coin, and the bell rings. What are the odds that the coin I flipped turned up heads?

                                      Now imagine that if it's heads there is a 100% chance the bell will ring, and if it's tails there's a 200% chance that the bell will ring (200% meaning 100%, then forgetting, then 100% again).

                                      [–]wiphiadminΦ 1ポイント2ポイント  (0子コメント)

                                      If it's helpful, here's a relevant /r/philosophy discussion from an earlier Wi-Phi post: https://www.reddit.com/r/philosophy/comments/2xhaxs/epistemology_the_sleeping_beauty_problem/

                                      [–]Oznog99 1ポイント2ポイント  (0子コメント)

                                      Hmm let's state it this way.

                                      We roll this game 100 times. Pure chance wins out and 50 come up Heads, 50 Tails. But Sleeping Beauty is woken 150 times because of the double-waking on Tails.

                                      If she consistently places $1 bets on Heads 150 times, she will have won 50 times and lost 100 times, for a net loss of $50.

                                      If she consistently places $1 bets on Tails 150 times, she will have won 100 times and lost 50 times, for a net win of $50.

                                      One could also say "I am going to flip a coin, if it is Heads I won't do anything. If it is Tails, I will ask you what chance you think the flip is Tails." When asked, you should answer "It's Tails. 100% chance. Final answer, it's Tails. Every time you ask me." You do not need to see him flip and fail to contact you. If those are the rules, it's 100% chance to be Tails when contacted because the result has been cherry-picked by the rule.

                                      So the next question: would a bookie give her 50/50 odds? How is it the bookie lose $50? He wouldn't- well, he shouldn't. Although she does not KNOW for certain that the coin is Tails, she's going to bet twice only if it's Tails. From the bookie's perspective, the first night is "fair", 50/50. Allowing her to wake up a second night and bet again only if it's Tails is bullshit for the bookie.

                                      One might as well ask a casino "if the roulette wheel gives me Black, I would like to automatically place a second bet of the same color and dollar value, which, full disclosure, I'm gonna bet on Black from the start every time!" The casino would say "no, you've broken the rules at that point. You get to have 2 winning bets if it is Black, and only one loss if it's Red? The second bet is always 100% certain to be Black, and a win for you."

                                      [–]goomyman 1ポイント2ポイント  (0子コメント)

                                      Here is the solution:

                                      Sleeping beauty wakes up and guesses Heads everytime.

                                      Is she right 50% of the time or 1/3rd of the time.

                                      [–]danisaacs 9ポイント10ポイント  (103子コメント)

                                      It's 1/2. The odds of a coin flip are always 1/2. Because Math.

                                      [–]Jonathan_Livengood 5ポイント6ポイント  (17子コメント)

                                      It's true that the unconditional probability of heads on the coin flip is 1/2, since the coin is fair. But the problem before us is to say something about the conditional probability. And the conditional probability might or might not be 1/2.

                                      Suppose we have two coins and both of them are fair. If I tell you that one of the coins came up heads, then the probability that the other came up heads is 1/2. So far so good.

                                      But now, suppose that we have a system such that if both coins come up heads, a bell rings. Suppose we observe one of the coins to be heads, but the bell does not ring. What probability should we assign to the other coin being heads? Seems that the conditional probability of heads is 0, not 1/2.

                                      The question in the Sleeping Beauty problem is whether or not Beauty has information -- coming from knowing the set-up of the experiment and knowing that she has been awakened -- that makes the conditional probability that the coin is heads in her specific circumstances different from the unconditional probability.

                                      EDIT: If you are tempted, like sexahol, to think that there is no reason aside from "semantic masturbation" (which is eponysterical), then I would very much like to gamble with you. ;)

                                      [–]PM_ME_KIND_THOUGHTS 10ポイント11ポイント  (38子コメント)

                                      If you have substantial circumstantial evidence that somebody committed a murder, and somebody asked what the odds are that they did it, would you say 50/50 because the only choices are that he either did it or didn't do it?

                                      That's what you are saying here.

                                      Sleeping beauty has circumstantial evidence that leads her to believe that she should NOT weigh the odds as a simple 50/50.

                                      [–]likesleague 28ポイント29ポイント  (25子コメント)

                                      Nope. Sleeping Beauty has incredible universal evidence (i.e. fair coin) that the odds are exactly 50/50.

                                      When the coin is heads (50% chance) Beauty wakes up once. When the coin is tails (50% chance) Beauty wakes up twice. The result of the coin "causes" the number of wake ups. It is not true in reverse. The wake up does not "cause" the result of the coin. If Beauty were to use such logic (i.e.: I've woken up, what does this tell me?) she would have to remember that though a tails causes two wake ups, it causes a single chain of wake ups. The wake ups are not independent of each other, and you do not add their probabilities. If you tried to think of it as adding probabilities (which is technically incorrect, but if you slightly altered the problem to include Beauty guessing what day she woke up on, and whether it was heads or tails, then you could consider each wake up effectively independent) then you'd say that p(heads and monday) = 50% (because there's a 50% chance of heads and a 100% chance of waking up on monday if heads is tossed) and p(tails and monday) = 25% (because there's a 50% chance of tails and a 50% chance that when she wakes up on a tails flip it's on monday) and p(tails and tuesday) = 25% (same reason as p(tails and monday)). Point is; if you're going to add probabilities based off of waking up, you need to remember that each wake up considered individually is weighted. The heads wake up is 50% and the others are 25%.

                                      [–]PM_ME_KIND_THOUGHTS 6ポイント7ポイント  (17子コメント)

                                      you are looking at it wrong though, I think.

                                      If you are sleeping beauty and you wake up, 2/3 of the time you will be on a tails wake up. Sleeping beauty has no way of knowing if she is in a chain of wake-ups caused by a tails flip or not, so she can't just include that in her "in the moment" calculations. every single wake up is its own thing.

                                      If she is straight up asked to guess if the coin was heads or tails, in a practical sense she should assign a probability of 2/3 to a tails outcome in order to make a better educated guess.

                                      probability is only useful because it helps us predict things. As I said in another comment, if your "correct" answer does not help sleeping beauty and mine can, then how can you call yours correct?

                                      [–]likesleague 16ポイント17ポイント  (6子コメント)

                                      She is asked the question, "What do you think is the probability that the coin came up heads?"

                                      If we're going to really understand the disagreement between 1/2ers and 1/3ers, we have to understand that the question is nonsense.

                                      probability is only useful because it helps us predict things

                                      Perhaps not to "help us" but yes, probability predicts things. You don't use probability to determine past events, you use probabilistic reasoning. If I drink a can of soda on 50% of days and I ask you what's the probability that I drank a can of soda today, you'd say 50%. That's the right answer to the slightly-rephrased Beauty question "when/before the coin was tossed, what was the probability that it would come up heads?"

                                      However, if I have a soda can in my head, you can use the relevant information to guess that I 100% had a soda today. However this does not mean that the probability that I would have had a soda today is anything different than 50%. Saying, "seeing as you're drinking a soda, the probability that you had a soda today is 100%" is backwards logic. I can use the fact that I drink soda 50% of days to create a probability (50% in this case) that I'll observe myself drinking soda on any given day, but I cannot use the information that I am drinking a soda to determine the probability that I drink soda on any given day. Such logic is basically the math equivalent of a post-hoc fallacy in an argument.

                                      To summarize:

                                      If Beauty was asked the question "[given that you know you have woken up] how likely is it that this wake up is a result of the coin coming up heads?" then 1/3 would be the obvious answer.

                                      If Beauty was asked the question, "what is the probability, upon being tossed, that the coin would have come up heads?" then 1/2 would be the obvious answer.

                                      The question, "What is the probability that the coin came up heads" is, in the context of the wake ups, misleading, nonsense, and very vague.

                                      [–]PM_ME_KIND_THOUGHTS 8ポイント9ポイント  (5子コメント)

                                      I think you are right in saying the question as presented in the video is vague and that's why there is the big argument.

                                      but I think, unless you say the whole experiment is just a ruse to trick us, the intent of the question is clear.

                                      wikipedia phrases the question as:

                                      "What is your belief now for the proposition that the coin landed heads?"

                                      some random princeton website phrases it as:

                                      When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?

                                      both of which point to the answer being, I think, pretty obviously: 1/3 chance of heads.

                                      [–]likesleague 2ポイント3ポイント  (4子コメント)

                                      I agree, though something feels off about this. I'm currently working on solving a problem about the probabilities of 1-information variables (call it M) in 2-information variable systems (M and N) in which M is correlated with N and N with M, but M is not correlated with M. In terms of probability, the Sleeping Beauty problem as phrased by Wikipedia/Princeton is a similar kind of system. If I work anything out on it, I'll try to remember to come back here and give the best explanation I can.

                                      [–]kiradotee 2ポイント3ポイント  (0子コメント)

                                      If I work anything out on it, I'll try to remember to come back here and give the best explanation I can.

                                      Ah, how can I subscribe to OP deliveries?

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                                        [–]EreTheWorldCrumbles 3ポイント4ポイント  (0子コメント)

                                        There is a simple solution to this problem, and it centers on being quite specific about what you are actually asking. If you are specific in your question, then you can be specific in your answer.
                                        These are the possible questions you could ask given the scenario and the clear solutions to those questions. No paradox or ambiguity necessary:

                                        question 1: "What must sleeping beauty choose in order to maximize the number of correct guesses.
                                        Answer: Tails. The relevant metric is the number of guesses uttered, not the number of coin-flips.

                                        question 2: "What must sleeping beauty choose in order to maximize the number of coin flips guessed correctly?
                                        Answer: It doesn't matter, as the odds are always 50/50. The relevant metric is the the number of coin flips, not the number of guesses.

                                        (The only contradiction in this second question comes into play if on monday and tuesday she makes two different guesses. That would make the coin flip null, as she was both right and wrong in her guess.)

                                        [–]johncheswick 2ポイント3ポイント  (0子コメント)

                                        2/3 of the time you will be on a tails wake up

                                        This is the fallacy in your argument. It is not 2/3 of the time, it is 2 out of 3 outcomes. Each of the three states does not suddenly have an equivalent chance of having occurred simply due to the fact that she woke up.

                                        Sleeping Beauty should be able to work in reverse and reason that something that is less likely to have happened could still have happened, ie. being woken up on Tuesday.

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                                          [–]drukath 0ポイント1ポイント  (0子コメント)

                                          Let's say I told you I was going to toss a fair coin and on a heads I will punch you in the face and on a tails I will give you $100.

                                          I toss the coin in secret and then give you $100.

                                          What is the probability that the coin came up heads?

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                                            [–]ddrddrddrddr 5ポイント6ポイント  (11子コメント)

                                            If she is supposed to maximize her chance of getting the toss right, she should answer tails because she gets two guesses if it is tails. The probability to begin with for heads or tails is still 1/2. So we're not changing probability, we're only taking advantage of having double the chances to answer.

                                            [–]Flaisse 4ポイント5ポイント  (21子コメント)

                                            There are three possibilities.

                                            She is woken on monday, and will not be woken again.

                                            She is woken on monday, and will be woken on tuesday

                                            She is woken on tuesday, and has no knowledge of the monday awakening

                                            From this, there is a 2/3 chance that the coin toss was tails.

                                            [–]AggregateTurtle 3ポイント4ポイント  (9子コメント)

                                            because of the amnesiac couldn't one argue it doesn't matter which of the 2 resulting days beauty is woken, they are the same track/subset of results, there is no seperate state there, only one state that will be run to completion, originating at the original 1/2 chance.

                                            [–]Flaisse 1ポイント2ポイント  (4子コメント)

                                            Think of it this way. What are the scenarios in which she awakens? There are three separate scenarios for this, and they are all equally likely to sleeping beauty, since she has no information to influence her decision.

                                            If all three scenarios are equally likely, the probabilities are 1/3 and 2/3.

                                            [–]SuperFunBot 3ポイント4ポイント  (2子コメント)

                                            But the three scenarios are not equally likely. There is a 50% chance of waking on Monday after heads, and 50% chance of waking either Monday or Tuesday after tails; (Maybe this means 25% chance of waking on Monday after tails, and 25% chance of waking on Tuesday after tails? But I'm not so sure of that.)

                                            Sleeping Beauty may find herself thinking, "It could be any of the three possible situations that I am waking up to." However, two of those situations have the same origin. She will never wake up on Tuesday if the coin flipped heads. She will always wake up Monday, regardless of the coin flip.

                                            [–]Flaisse 1ポイント2ポイント  (1子コメント)

                                            sure, but we're not looking at this in a linear sense. We know that she is woken on monday, so to us when she wakes on monday its a 1/2 chance of either flip. But to her, the day isn't known, so she has 3 possible outcomes.

                                            Picture this. Heads, she gets woken once on monday. Tails, she is woken a thousand times, being amnesic to all the other awakenings. Do you still think that there's a 1/2 chance of it being heads when she is woken?

                                            [–]SuperFunBot 2ポイント3ポイント  (0子コメント)

                                            Yes. To say that there are three possibilities and therefore 33% chance of each is the same as saying that this lotto ticket will either win or lose, so it's 50/50.

                                            Waking on Monday after heads and waking Tuesday after tails are not equally likely. There is only a 50% chance that it is possible for her to wake on Tuesday.

                                            What would change it for me is if the experiment were to be performed many times, and she knew ahead of time that she would be asked every time she woke up whether it had been heads or tails. She would have a reasonable expectation that over time the total number of times waking up after tails would be double the total number of times waking up after heads, therefore she would maximize her correctness by always guessing that it had been tails. But that is not the problem as presented. As I describe the problem, tails has an advantage because she frequently gets two points for a single flip if she always guesses tails. As the video describes the problem, there is no bonus for the multiple opportunities to wake up after tails.

                                            [–]Blackdragonproject 0ポイント1ポイント  (0子コメント)

                                            But they are not all equally likely to sleeping beauty. The probabilities of those three results are 1/2, 1/4, and 1/4. When she wakes up, she knows that the probability of it being a Tuesday after the coin flipped tails is 1/4, not 1/3.

                                            [–]It_stands_to_reason 0ポイント1ポイント  (3子コメント)

                                            I agree. /u/Flaisse, in his reply to you has stated that those are three separate scenarios, but it seems that the second and third outcome or really the same. From either the experimenter or SB's perspective, there are only two outcomes. We could arbitrarily throw in many stages for the second (and I guess the first) outcome (I'll wake you up, give you a drink of water, change the sheets), but it doesn't change the fact that in the end she will wake up from one of two series of events.

                                            [–]Flaisse 2ポイント3ポイント  (2子コメント)

                                            They are the same. But we're working on Beauty's perspective. There are 3 scenarios, and in her point of view, none are more likely.

                                            She needs more information, such as the tester telling her it is monday or tuesday to narrow her options.

                                            [–]johncheswick 1ポイント2ポイント  (0子コメント)

                                            She has more information. She knows a 50 / 50 coin flip led her to her scenario.

                                            She can work backwards and deduce that there is a chance that it is Tuesday, but that the odds that it is Tuesday are not the same as the odds that the coin landed on heads.

                                            [–]It_stands_to_reason 0ポイント1ポイント  (0子コメント)

                                            I can understand that reasoning and obviously this problem exists because it creates paradoxical solutions, but I disagree, personally, that there are three possibilities. As you said, she lacks the information to narrow her options. Thus, the only information she has is that a fair coin was flipped.

                                            The alternative point of view leads us down a rabbit hole, where the experimenter could give her amnesia on Tuesday and wake her on Wednesday, or after a week. Those extra days wouldn't have affected the outcome of the coin, because the coin was flipped the first day.

                                            [–]danisaacs 0ポイント1ポイント  (0子コメント)

                                            I confess I didn't listen to the whole thing. How many times is the coin flipped?

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                                              [–]itisike 1ポイント2ポイント  (0子コメント)

                                              So if I see the coin land on heads, I should still think there was a 50% chance that I imagined it, because math? Or, perhaps, evidence can change probabilities.

                                              To give a more intuitive case: I lock SB in a room and put her to sleep. I flip a coin. If heads, I let her out. If tails, I pick a random number from 1-100 with a rng, and for all numbers except 100, I kill her in her sleep. If I picked 100, I wake her up. You are SB, you wake up. Does your "math" say that the probability of heads was equal to tails? This is pretty much the negative version of the problem.

                                              Or as I've said elsewhere in this thread, consider her posteriors after learning that it's Monday. She can't still think heads and tails are equally likely, but you would naively claim she still should.

                                              [–]TychoCelchuuuΦ 0ポイント1ポイント  (2子コメント)

                                              Imagine that you are in a room and I am in another room, flipping a coin over and over. Every time the coin comes up heads, I text you a picture of the coin. Every time the coin comes up tails, I make a check-mark on a piece of paper, and whenever I've made ten check marks, I text you a picture of the coin. So in effect for every 10 text messages with a picture of the coin having come up heads, you also get 1 text message with a picture of the coin having come up tails. Of course, the messages aren't evenly spaced: sometimes you get 9 heads messages then a tails message, sometimes you get 6 heads then a tails, etc.

                                              Now imagine someone else comes into the room and pulls a gun on you. They say "I want you to predict whether the next message is going to have a picture of the coin showing heads or the coin showing tails. If you get it wrong I shoot you."

                                              One thing you could say is that "well every time Tycho flips the coin, it's 50/50, so whatever message I get, that coin is a 50/50 heads/tails coin, so I'll just make a random guess. Because Math."

                                              But that would be stupid. Surely you ought to guess heads. It's more likely that you'll get a message about a heads coin than a tails coin.

                                              This illustrates the point at issue here: if you only know that a coin is being flipped, then you would just make a 50/50 guess. But if the results of a coin flip determine the information you get about the flip, then sometimes this can change what you will guess.

                                              Obviously in the sleeping beauty case the rules are more convoluted than "tell you about all the heads but only 1 in 10 of the tails." But we could recreate the sleeping beauty rules with text messages. If we did so, I think you would see why it would make sense to guess (under penalty of death) something other than 50/50.

                                              [–]danisaacs 0ポイント1ポイント  (1子コメント)

                                              Well, that's a different scenario. The odds that an image will be H or T (and NOT the coin flip itself) are most certainly not 1/2.

                                              Perhaps I did not listen carefully enough to the problem. What information does SB have?

                                              [–]The_Yar 1ポイント2ポイント  (7子コメント)

                                              What if the coin flip already happened, and you're looking at it and you see that it's heads?

                                              Would you still insist that there's a 50% chance it's tails?

                                              [–]danisaacs 0ポイント1ポイント  (6子コメント)

                                              No, but you've seen the outcome. How does SB, at the moment she is making a guess, know anything relevant?

                                              [–]The_Yar 0ポイント1ポイント  (4子コメント)

                                              She knows she's just been woken and asked a question, rather than sitting at home drinking tea.

                                              She knows today might be Tuesday, and if it is, the flip must have been tails.

                                              [–]danisaacs 0ポイント1ポイント  (3子コメント)

                                              Ok, so given what she knows, the odds are 1/2. She doesn't know what day it is. That is why I don't think this is a conditional probability, she doesn't have any information that would change her odds.

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                                                  [–]danisaacs 0ポイント1ポイント  (0子コメント)

                                                  Presuming a fair coin and no environmental influences, of course. :)

                                                  [–]ifthisdoesntfitillki 0ポイント1ポイント  (0子コメント)

                                                  This isn't asking about the odds of a coin flip, it's asking about the odds of the coin flip in this scenario.

                                                  [–]Readitwhileipoo 0ポイント1ポイント  (1子コメント)

                                                  Based on that logic I should purchase my lottery ticket asap

                                                  [–]danisaacs 1ポイント2ポイント  (0子コメント)

                                                  No, those odds are much worse than 1/2. Also because of math.

                                                  [–]xanbo 0ポイント1ポイント  (0子コメント)

                                                  The question really isn't "what are the odds of a coin flip coming up heads?" It is "what are the chances you live in a reality in which the coin toss came up heads?" Those are two very different questions.

                                                  [–]Vicker3000 0ポイント1ポイント  (0子コメント)

                                                  Let's say I flip a coin. It comes up heads. I show it to you. You see that it came up heads. I now ask you, "What is the probability that that specific coin toss came up heads?" You would answer that you are 100% certain that that specific coin toss came up heads.

                                                  [–]drukath 0ポイント1ポイント  (2子コメント)

                                                  The chance of a future event occurring is 1/2.

                                                  The chance of a past event that occurred having occurred is 1. So the coin toss is no longer 1/2 because it happened and must be one or the other it cannot be in a state of uncertainty.

                                                  Therefore the chances of you being able to say what occurred is based upon the information you have of the event, not on the chance of a future coin toss.

                                                  The experimenter, having seen a heads, would not say that chance that the coin toss was a heads was 1/2. He's seen the result. He'd say it was definitely, 100%, a heads that was tossed.

                                                  Bayes proved this a couple hundred years ago. Because maths.

                                                  [–]danisaacs 0ポイント1ポイント  (1子コメント)

                                                  No, until you know the outcome, the odds are still 1/2. The cat is both alive and dead, until you know one or the other, to throw in a red herring from another discipline. SB has no information that would change her odds of being right.

                                                  [–]drukath 0ポイント1ポイント  (0子コメント)

                                                  No, that's completely incorrect. Conscious observation does not drive reality. It is not in a state of flux until you look, and the Schrodinger's cat experiment is often misquoted in this way.

                                                  For example the experimenter does know what the result is. The experimenter also knows what day it is when he wakes SB. It is absurd to say that the day could change based upon SB knowing or not knowing the result. The rest of reality doesn't get reset to Monday based upon SB learning that it is a Monday.

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                                                    [–]_GeneParmesan_ 2ポイント3ポイント  (0子コメント)

                                                    Utterly shit video

                                                    Invalid question, completely flawed. Nothing is defined correctly, it's entirely and utterly shit and highlighting how people cannot define problems.

                                                    Utterly shit.

                                                    [–]chinchalinchin 3ポイント4ポイント  (5子コメント)

                                                    No one has yet cracked this problem, from what I can tell.

                                                    The problem is that the scenario under consideration does not satisfy the axioms of probability. There are three events taking place: waking up from heads (W1) , waking up from tails the first time (W2), waking up from tails the second time (W3). However, the sample space is only one element big: heads. There are only two possible subsets of the sample space: heads or its exclusion, not-heads (corresponding to tails). There is a many-to-one relationship between the events measured and the sample space (If you want to get deep into mathematical jargon, technically, we should say it is a many-to-one relationship between the events and the power set, sometimes called the omega set, i.e. the set of all subsets of a space). In other words, the events being observed cannot be measured with given sample space, because there is no way to uniquely specify events with a degenerate sample space.

                                                    From my undergraduate level probability textbooks: "Let S be the sample space of a random phenomenon. Suppose to each event A of S (A being a subset of S), a number denoted by P(A) is associated with A. If P satisfies the following axioms...yadda yadda yadda..."

                                                    The problem here is the events A {W1, W2, W3} are not a subset (or a function of the subsets) of the sample space S {H}. Therefore, probability cannot say anything about this problem. In order for a calculus like probability theory to be satisfied, the problem must first meet the conditions of probability-theory.

                                                    Consider the abstract system, 1. there exist exactly three distinct Fe's in this system. 2. Any two distinct Fe's belong to exactly one Fo. 3. Not all Fe's belong to the same Fo. 4. Any two distinct Fo's contain at least one Fe that belongs to the both.

                                                    This system of assumptions prescribes a theorem-structure. For instance, it can easily be shown: "Two distinct Fo's contain exactly one Fe." or "There are exactly three Fe's."

                                                    Any problem that meets these assumptions will, by necessity of its form, have these theorems as a result. For instance, if we interpret Fe's as people and Fo's as committees, it makes perfect sense and thus the theorems are valid. If, however, Fe's are taken to be as, say, books and Fo's are taken to be shelves, the assumptions are not met and theorems of the system do not apply.

                                                    Analogously, people are trying to apply the theorems of probability to this problem, when this problem cannot be analyzed with probability without more information. There is no 'probability' associated with the event of waking up, since there are three states of waking up which cannot be measured in binary terms.

                                                    ... ... ...

                                                    This is why you shouldn't let philosophers do math.

                                                    EDIT: Minor grammatical errors and further clarification.

                                                    [–]WilliamKiely[S] 1ポイント2ポイント  (4子コメント)

                                                    What you said explains why it's a philosophy problem, not a math problem. Upon waking up during the experiment, what credence ought Beauty assign to the proposition that the coin came up heads? I think the 1/3 answer is correct.

                                                    [–]chinchalinchin 1ポイント2ポイント  (2子コメント)

                                                    You say it is not a mathematical problem then invoke the term 'credence' as if it is meant to imply anything other than 'probability'. You even go so far to say this credence is 1/3. How is one to intrepret a 'credence' of 1/3? What you mean is probability. And there is no probability that can be assigned to her state, for the aforementioned reasons. What would that '1/3' even mean? You would be saying there is a 1 in 3 chance she is in some state, but this says nothing at all about the coin being heads or tails.

                                                    In effect, her state fails the 'vertical line test', if you recall that from algebra or pre-cal. Her state has no functional dependence upon the coin. You are tricked into thinking it does because there is a causal connection. If you wanted to treat this problem with probability, you would have to assume there is no difference in the two states produced by tails. This goes against our intuition since there is a causal, temporal connection between the two events. However, probability theory cannot deal with violation of its assumptions. This reduncancy with respect to tails and her state prevents us even interpretting the problem in terms of probability.

                                                    [–]WilliamKiely[S] 1ポイント2ポイント  (1子コメント)

                                                    You're right that I was wrong when I said that it's not a math problem. It definitely involves math/probability. What I meant is that it's not only a math/probability problem, but also is a philosophical problem (specifically an epistemological problem). The observation selection effects (Bostrom's term) at play bring it into the realm of philosophy. Yes, it still involves math/probability.

                                                    You would be saying there is a 1 in 3 chance she is in some state, but this says nothing at all about the coin being heads or tails.

                                                    Yes it does, because the coin came up heads if and only if she is in the "Monday-Heads" state. If she ought to believe there is a 1/3 chance she's in that state (which I believe is correct), then she also ought to believe that there is a 1/3 chance that the coin came up heads.

                                                    EDIT: I think this reasoning for 1/3 is sound: https://www.princeton.edu/~adame/papers/sleeping/sleeping.pdf

                                                    EDIT 2: Also note that this paper replying to the above paper and arguing for the 1/2 view is mistaken. Specifically, I believe the premise L1 is wrong. When I read it I initially thought it was true, but once I followed through the logic to L6, I felt confident that L6 was wrong.

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                                                      [–]4thShell 4ポイント5ポイント  (1子コメント)

                                                      It is 1 or 0, because the time she was asked the question, it already happened. ( ͡° ͜ʖ ͡°)

                                                      [–]drukath 0ポイント1ポイント  (0子コメント)

                                                      This! But only to people that know the result. So to the experimenter it is 1 or 0.

                                                      To SB she cannot be that certain, so it is (1/3 * 1)+(1/3 * 0)+(1/3 * 0) = 1/3.

                                                      [–]dr3i_ 1ポイント2ポイント  (3子コメント)

                                                      It doesn't matter what Sleeping Beauty thinks - she has no idea whether she is waking up for the first or 100th time. You wouldn't ask HER for the results of the coin toss in this experiment.

                                                      The coin is still 1/2 heads and 1/2 tails.

                                                      Sleeping beauty's perception of this whole experiment and the answer to the misleading question at the end from her perspective is 1/3.

                                                      The chances of the coin remain 1/2.

                                                      The chances of Sleeping Beauty using logic to guess correctly heads or tails in 1/3.

                                                      [–]ddrddrddrddr 1ポイント2ポイント  (8子コメント)

                                                      Okay this is not a philosophical issue, it's just conflation of two different issues. There are two different issues being considered here. First issue is what is the probability that a coin is head or tails is. The second is what the probability she is waking up to a head or tail is. The first has a 50/50 and the second is 33/66. Just because the answers are different doesn't mean they some how conflict.

                                                      [–]WoodyTwigs 0ポイント1ポイント  (1子コメント)

                                                      What's the reasoning behind all this?

                                                      [–]thatmuppet4 0ポイント1ポイント  (6子コメント)

                                                      there are three solutions, but the ones with tails having been flipped are split in an absolute 50/50 chance. The probability of each tails possibility would be 1/4 because they are contingent on the first flip being tails, so the the answer should be 1/2.

                                                      [–]ifthisdoesntfitillki 3ポイント4ポイント  (5子コメント)

                                                      That's not correct. If you think about it, you will realise heads Monday, tails Monday and tails Tuesday all have an equal probability of being the day of her awakening. Remember, if tails Monday occurs then tails Tuesday is absolutely guaranteed to follow. Therefore the tails days do not have a 1/4 chance of occurring.

                                                      [–]gpc 0ポイント1ポイント  (1子コメント)

                                                      For tails, it's not the "chance of occurring" what matters, but the probability of being in a specific moment in time. The fact that tails Tuesday is guaranteed to follow tails Monday is irrelevant to that question. The coin flip generates a 50/50 situation. Now, if tails happens, when she wakes up there's 1/2 chance it's Monday and 1/2 chance it is Tuesday; that means 1/4 chance for each in the whole scheme of things.

                                                      [–]ifthisdoesntfitillki 1ポイント2ポイント  (0子コメント)

                                                      No, that's wrong. If the Tuesday is guaranteed to follow the Monday then you will NOT have a 1/4 chance of awakening on either of the tails days. This is not a case of "1/2 chance of tails and two tails days thus 1/4 and 1/4" because the two events are not exclusive. The chance for either tails day being 1/4 would be true if you could only wake up on EITHER a tails Monday OR a tails Tuesday, but never one and then the other (without knowing, which is what adds nuance to the question)

                                                      All three days have an equal probability of occurring. Think about it: heads Monday occurs 5/10 times. Tails Monday occurs 5/10 times. If tails Monday occurs, so will tails Tuesday. Over these ten iterations, you will be awoken 15 times, 5 times on each day.

                                                      [–]galosheswild 0ポイント1ポイント  (1子コメント)

                                                      He's not saying the tails days have a 1/4 chance of ever occurring, but rather they have a 1/4 chance of presently occurring. Half of the time they will both occur, and if you were to guess "I have been awoken on exactly Monday and it came tails" then you would be correct 1/4 times (that is what he is stating at least).

                                                      [–]ifthisdoesntfitillki 1ポイント2ポイント  (0子コメント)

                                                      I know that is what he is stating and he's wrong

                                                      [–]t3nk3n 0ポイント1ポイント  (0子コメント)

                                                      1/3 of the time that Sleeping Beauty wakes up, I am eating a green apple. Given that Sleeping Beauty has woken up, what is the probability that I am eating a green apple?

                                                      How exactly does the percentage of apples I eat that are green even factor into this question?

                                                      [–]mirroredfate 0ポイント1ポイント  (1子コメント)

                                                      My intuition tells me this is really two separate problems.

                                                      First, there is the probability that the coin will come up heads vs tails. This is always a 50/50 scenario. There is always a 1/2 probability the coin will come up heads (or tails).

                                                      The more subtle issue is that of Sleeping Beauty's (SB) correctness. We assume that SB wants to be correct in her answer, so which answer, then, gives her the highest probability of correctness?

                                                      This is where I find it difficult to explain. Assuming SB is perfectly rational, she knows that she should answer tails every time to achieve the highest accuracy. If she answers tails, her accuracy looks like this:

                                                      [T,T,T] = [F,T,T].

                                                      which is 2/3 accurate.

                                                      But if she answers heads, it looks like this:

                                                      [H,H,H] = [T,F,F]

                                                      which is only 1/3 accurate.

                                                      If she decides to guess, it would look like one of the following:

                                                      [H,T,T] = [T,T,T]
                                                      [H,T,H] = [T,T,F]
                                                      [H,H,T] = [T,F,T]
                                                      [T,H,T] = [T,F,T]
                                                      [T,T,H] = [F,F,T]
                                                      [T,H,H] = [F,F,F]

                                                      So it is interesting to note that these average out to 50/50- for any given guess, she has a 50% chance of being accurate.

                                                      Clearly, then it is in her best interest to always guess tails, because although it will not alter the probability of the coin toss, it will impact the number of times she will guess correctly.

                                                      Does that make sense?

                                                      [–]servant91 0ポイント1ポイント  (0子コメント)

                                                      What's the reasoning behind all this?

                                                      [–]Mikeg90805 0ポイント1ポイント  (2子コメント)

                                                      I don't get the thing about "heads world" the million and one thing can someone explain that to me

                                                      [–]Deviver 0ポイント1ポイント  (1子コメント)

                                                      Think of it as not the probability of the coin. But the probability of what happened.

                                                      In the event the heads was flipped she would be woken up on Monday and asked this question

                                                      In the event of tails. She would be waken up on Monday and asked this question.

                                                      Then she would be drugged to forget and woken up on Tuesday to be asked this question

                                                      If you repeat the druged and woken up part a million times, the argument is that if you consider each one of the awakenings as a possibility which one are you in. Since heads only lead, to one of the possible awakenings and tails leads to the other the probability (following the 1/3 argument) would be 1/(infinity +1)

                                                      [–]fellow_earthing 0ポイント1ポイント  (1子コメント)

                                                      Maybe another slightly weirder way to state the halfer position is that there are actually four possible equally-likely outcomes: Heads Monday(A), Heads Monday(B), Tails Monday, and Tails Tuesday, and that Heads Monday A and B are simply indistinguishable from one another, type identical, but not numerically identical, and each occurring with 1/4 frequency.

                                                      Since SB knows the rules of the game prior to the toss, why would she choose to see it any other way afterwards? What new information is she gaining upon being asked? Why is it suddenly more likely that she's inhabiting one of the many small-sized "tails" realities, rather than the big-sized "heads" reality?

                                                      [–]elbruce 0ポイント1ポイント  (0子コメント)

                                                      She should describe a curve. 50% it's Monday, 12.5% it's Tuesday, 6.75% it's Wedneseday, and so on. But she can't give an answer, just the curve.

                                                      [–]bababouie 0ポイント1ポイント  (0子コメント)

                                                      This is messed up. The results are skewed in 1/3rd scenario because they are making it seem that you can ask more than once of the probability of it being heads...meaning they are saying they can ask twice on Monday, and once on Tuesday which skews the results.

                                                      If they only can ask once after she wakes up, then the answer will always be 1/2 regardless of what day they ask her in the Tails scenario. By asking more than once it is as if you are reflipping, but by framing the scenario as being Tuesday and on the tails side...you have dictated the result being Tails.

                                                      In other words, flip the coin 10 times and ask 10 times (once per flip) results in 1/2 every time. Flip the coin once, and asking 10 times is essentially creating 9 tails results and 1 heads result which you are artificially creating by basis of your scenario. Essentially you have dictated the results and created 10 events. Each time you ask in a one flip scenario, you're adding a dictated result to the tails side (regardless of # of days that she could be sleeping).

                                                      I would love to hear an explanation of why this isn't right.

                                                      [–]scullian3689 0ポイント1ポイント  (0子コメント)

                                                      Maybe another slightly weirder way to state the halfer position is that there are actually four possible equally-likely outcomes: Heads Monday(A), Heads Monday(B), Tails Monday, and Tails Tuesday, and that Heads Monday A and B are simply indistinguishable from one another, type identical, but not numerically identical, and each occurring with 1/4 frequency.

                                                      Since SB knows the rules of the game prior to the toss, why would she choose to see it any other way afterwards? What new information is she gaining upon being asked? Why is it suddenly more likely that she's inhabiting one of the many small-sized "tails" realities, rather than the big-sized "heads" reality?

                                                      [–]completely_ordinary 0ポイント1ポイント  (0子コメント)

                                                      I don't get why people think it could be 1/3, let's say that if the coin flip was tails she would go to sleep and wake up one million times like in the video. This still has no effect on what the coin flip probability is. The coin flip and the fact that the beauty wakes are completely different events that occur in different time slots. As long as there is an initial coin flip she is going to wake up. Doesn't matter when (which day of the week it is) She won't remember that she woke up the day before anyways. There is no reason behind saying that it should be 1/3 , 1/4 or anything other than 1/2, because she will wake up not remembering anything regardless of what the result of the coin flip is. you think of these things as completely sequential events, First there is a coin flip and after that the beauty wakes up, doesn't matter when she wakes up. That's why i say it's 1/2. You can't think of this as observing her over time, say checking up on monday and then tuesday. It's said that she doesn't remember so neither should you. You shouldn't look at it trying to observe it over time but rather just observe it on that day only. If you keep your observation over time, then you might say 1/3 , 1/4 etc. But i don't think that it's right since she doesn't remember herself everytime she wakes how many times she woke up before. (every day is a clean start :D )

                                                      [–]Alljay_Everyjay 0ポイント1ポイント  (0子コメント)

                                                      Gabby from OMGfacts?????

                                                      [–]shamrock771 0ポイント1ポイント  (0子コメント)

                                                      Could someone please explain a bit why, from the coins perspective, it would appear to have been tails a million times in a row. I didn't understand that bit at all!

                                                      [–]jollyrogerninja 0ポイント1ポイント  (0子コメント)

                                                      This scenario really needs to be remade into a game theory problem.

                                                      A team of 3 people are placed in three different rooms with no knowledge of each other or what's going on outside. Then, outside the rooms, a coin is flipped. If the coin is Heads, one person chosen randomly will be asked to guess the coin outcome. If the coin is Tails, two people chosen randomly will be asked to guess the coin outcome. Every correct guess gains the team a point.

                                                      In order to maximize the number of points earned, each player ought to guess "tails" if asked. If it is tails, the team will gain two points, if heads, the team will gain zero points. Suppose the team chose to always guess heads. If heads, the team gains one point, if tails, they gain zero points. It is advantageous to guess tails. The probability of being correct is still 50/50, but you gain more for being correct if it's tails. If you guess heads, you have a 50/50 chance of gaining one point, if you guess tails, you have a 50/50 chance of gaining two points.

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