Thank you for this! This is exactly my thoughts as well. The multiple wake-ups are not independent events. From her perspective she only wakes up once, since her memory is wiped. The *researchers* may be able to use the information of what day she is being awakened to deduce what the coin was, but even so that doesn’t change the probability of the coin flip.
If you know the day, then the conditional probability (given that you know it is Monday, or given that you know it is Tuesday) is indeed no longer 1/2. P(heads | Tuesday) = 0 P(heads | Monday) = 2/3 But SB is not given any information when she wakes up. She already knew that she was going to wake up whether heads or tails. It seems thirders tend to accuse halfers of ignoring the conditional part, but they themselves do not recognise that SB does not get any new information. She learns that she woke up, but the probability of waking up is 1 and she knows that.
@@ronald3836Exactly! Thirders believe that since the SB is awakened, there are more chances she is being awakened one of the two times because of a tails.
But she doesn’t have to know what day it is to have a 1/3 chance. We have Monday Heads, with a 1/3 chance (the 2/3 chance of it being Monday times the 50% chance of it being heads) Monday tails with a 1/3 chance (the 2/3 chance of it being Monday times the 50% chance of it being tails (if heads has a 50% chance and tails is the only other outcome, it also has a 50% chance)) Tuesday Tails with a 1/3 chance (The 1/3 chance of it being Tuesday times the 100% chance of it being tails (if heads has a 0% chance on Tuesday and tails is the only other outcome, it has a 100% chance)) All she knows is that she will wake up and the coin has a 50/50 chance per scenario, so if 2/3 scenarios favor the 50% that is tails, that would give heads a 1/3 chance. While you are right that she is not given any new information, she still knows that she is either waking up on Monday in 2/3 possible times to be woken up and she is being woken up on Tuesday 1/3 possible times to be woken up.
this can be easily simplified to the following model: imagine there are 100 people sleeping in one room, if the coin is heads then guy #1 is woken up, if it is tails, guys #2-100 are woken up. You are a random person being tested and have no knowledge of what number you are, then are asked the 2 following questions: 1. what WAS the probability of the coin being heads or tails?: it was 50% 2. now that you are awake, what do you think the result of the coinflip was?: you would be right answering tails 99 out of 100 times...
@@samuelhernandez4135 Same low ods of sleeping beauty waking up one time instead of 2. In this example "every wake up" event is a different person and all happens the same day, but ofc you could do heads and tails where SB wakes up 99 times with heads and 1 time with tails... it is just exagerating the probablities for better visualization, as in the monty hall problem
@@pscl_jsc This is my take on it too. Waking up several people gives you the counterargument for when halfers say "it is a fair coin, it is 50/50" as it is more than a fair coin: it is a fair coin AND sleeping beauty was woken up (which she would do twice on tails and only once on heads). However, I would love to see your response to this one: It is the same experiment but the coins is flipped in front of sleeping beauty on Sunday before she is put to sleep. They do not show her what it landed on but they ask her, what is the probability it was heads? She will obviously answer 1/2. Now, the experiment is carried out. When she is woken up, how is it logical for her to now say the coin has a 1/3rd chance of it being heads even though it is the same coin flip she previously called 1/2 chance of heads and she has been given no new information.
@@pscl_jscI can see your point, but I disagree. This model only works in a timeline that everyone exists together, and only one person is awakened. In the original question, a coin is flipped to determine if she would wake up in the heads timeline or the tails timeline. When she wakes up, there's still equal chance that she's living in the heads timeline as the tails timeline
@@pscl_jsc No, the odds are totally different. The person knew the rules going into it, and knew that there was a huge possibility they would not be woken up unless Tails. Being woken up was not guaranteed, and it added information.
I fully agree that there is no paradox here for mathematicians. (Apparently philosophers do find it difficult to understand probabilities.) But I have a question: Suppose you wake up and you are told it is Monday. What are now the probabilities?
@@christoffer1769 Incorrect 😀 P(tails) = 1/2, therefore P(tails and Monday) = P(tails and Tuesday) = 1/4. Since P(heads and Monday) = P(heads) = 1/2, this gives P(Monday) = 3/4, P(heads and Monday) = 2/3 and P(tails and Monday) = 1/3. (And this is at first sight surprising and perhaps "paradoxical".) So even though P(heads) always remains 1/2, the conditional probability of heads given what you know might be different. However, since SB is not given any new information when she wakes up (she already knew she was going to wake up whether heads or tails), P(heads | she woke up) remains 1/2.
@@ronald3836 I agree with your calculations, by saying that: P(Monday and Head) = 1/2, and P(Monday and Tail) = 1/4, Bayes law imply that P(Head | Monday) = 2/3. I would however say that we have found an expetion where Bayes law does not apply. In this case the right way to assign probabilities is to say P(Head | Monday) = 1/2, and P(Tail | Monday) = 1/2. Given that it is monday does not give you any information about the outcome of the coin, a "monday-awakening" will occur for any outcome, hence the coin-outcomes should still be given 50% probability each.
@@christoffer1769 by definition of conditional probabilities, P(head | Monday) = P(head AND Monday) / P(Monday) = (1/2) / (1/2 + 1/4) = 2/3. There is no exception to this.
@@ronald3836 I think this example shows that Bayes rule has exceptions. In this example two different types of uncertainties are mixed together in a strange way; the toin coss which is a "classical" random event, and then the SB's lack of knowledge (a completely different type of uncertainty) which forces her to divide her believe (i.e. divide P(Tail)=1/2 into P(Tail and Monday) =1/4 and P(Tail and Tuesday)=1/4)). It is no reason to believe Bayes theorem should handle this strange situation. And further, if one is a halfer and accept the "no new information"-argument, it would be contradictory to suddenly take a "thirder view-point" and assume monday gives you new information. Monday gives no new information about the coin toss, just as waking up gives no new information.
Your problem with an answer being reached by means of consensus is justified. Obviously, an incorrect answer is not made correct by receiving a majority vote. (With one exception… Democracy. You’d be surprised how many people will accept something as fact, just because most of the people around them voted as such.) However, taking issue with an answer being reached by consensus in this situation is irrelevant. He did not suggest that a correct answer would be reached by voting. He didn’t imply that there was even a decision to be made. He only asked for people to vote for which answer they liked or agreed with the most… that’s all. You assumed that he has an intended use for the votes, since he made no such claim. That being the case, the correct statement would be to say that you have a problem with your own assumption that he intends to base a claim of a conclusive answer on the votes he receives… Which is not a very good way to begin a rebuttal video… and we’re only 39 seconds into your response.
I kind of feel like most of the halfers' reasoning ultimately relies on "bro lmao it's a fair coin you *know* the chances are 1/2H and 1/2T. Stop confusing yourself with random bullshit", but man, *every* person who tackled the SB properly is smart enough to understand the meaning of "fair coin". We all get it. The unsettling thing about this problem is that bayesian reasoning *still* seems to suggest that, from the SB perspective, every time she's asked the question she is more likely to be in a "the coin landed tails" scenario. I'll basically copypaste the wikipedia page for this, since it's so well done. Let's assume that the sleeping beauty, when woken up, was *told* that the coin landed Tails (so it could be either Monday or Tuesday - she doesn't know, of course). Now a crucial part of the reasoning: would you agree with me that, in this scenario, to the eyes of the SB, it would be equally likely for the day to be either Monday or Tuesday? If the answer is yes, we can write down the following equivalence between conditioned probabilities: P(Monday|Tails) = P(Tuesday|Tails) That is, the probability that today is Monday (assuming we know that the coin flipped Tails) is equal to the probability that today is Tuesday (assuming we know that the coin flipped Tails). The conditioned probability of an event P(A|B) is basically just the probability of both A and B happening, but "renormalized" on the probability of B happening (because we know that B happened) : P(A|B) = P(A and B)/P(B). We deduce that P(Monday|Tails) = P(Monday and Tails)/P(Tails), so that in the equivalence found before we can simplify a P(Tails) to get: P(Monday and Tails) = P(Tuesday and Tails) We now reason similarly to before: let's assume that it was *told* to the SB that today is Monday (but she still doesn't know the outcome of the coin toss). Crucial assumption #2: would you agree that, in this scenario, to the eyes of the SB, it would be equally likely that the coin either flipped Heads or flipped Tails, since the Monday wakeup is scheduled in either case? If the answer is yes, we deduce: P(Heads|Monday) = P(Tails|Monday) Again, we simplify a P(Monday) to the denominator, and get: P(Heads and Monday) = P(Tails and Monday) But we also already found that P(Monday and Tails) = P(Tuesday and Tails) (of course, P(A and B)=P(B and A), since they both represent the probability of both events happening together), so that we deduce: P(Mon and H) = P(Mon and T) = P(Tue and T) In other words, _to the eyes of the SB, every time she wakes up these three events are all equally likely_ . Since in two of these equally likely scenarios the coin flipped Tails while only in one it flipped Heads, we deduce that the best answer of the SB based on its priors and assumptions is the thirder position. *Yes*, it is true that all the waking up events are not independent from each other - in fact, if you are woken up any day outside Monday the probability of Tails is 100%. But the whole point of the SB setup is that _she has no idea which day is today_ , nor how many times she's already been asked the question. This corresponds to a lot of information being lost, and if you run the maths and assume that Bayesian probability works, you end up in the thirder position. It's true that if it comes up Tails you already know that the SB will be woken up two times, but to her every awakening is a clear new experience: she has no way to distinguish neither of them from the single wake up caused by the Heads outcome, and she has to weight this in when making its guess on the outcome. This is reflected in the first crucial assumption I made: even if the SB was told that the outcome is Tails, thinking "Oh, it's easy! I know they'll wake me up two times in total!" doesn't give her full intel of her situation; she's still stuck with "But wait... Did they already wake me up, or is this the first time?". Let's put it this way. The thirder position is the correct answer to the question: "what is the probability that today, right now, the situation you're experiencing was caused by the fact that the outcome was Tails?", Which to me is very similar to the question: "What is the probability that the coin came up Tails?", maybe with a subtle "[given that you're awake right now?]" implied (which, together with the lack of information regarding the current day and past days, is the bit that causes all the trouble). If you accept the two assumptions that I labelled as "crucial" to be true, there's little escape. In a bayesian analysis of the problem, the Tails outcome simply weights more in the mind of the sleeping beauty - because the experiment conditions don't allow her to catch the blatant correlation between being woken up 2 times and the coin flipping Tails. Every day weights the same, since she doesn't know if it's a Monday or not.
She is asked what the probability is. From her perspective. The information that she does know is 1. Coin is flipped. 2. She is woken up on each day. That's all she knows. So the question from her perspective (the actual question). There are implications (as stated) like "she should". Why should anything change if she has no new info. From her perspective Tuesday is no different than Monday to her. And the question is "from her perspective".
You misunderstood the question. SB is asked about her credence that the coin came up heads (at a known past time). She is not asked about the expected behavior of a fair coin. So it's a problem in forensics: She must consider the evidence, in addition to the physical behavior of a flipped coin. The physical behavior gives her prior credence -- her expectation of the outcome of an arbitrary flip of the coin. Put in odds form, this is 1:1 for a fair coin. But she isn't asked about an arbitrary flip. She is asked about one particular flip, the one that happened Sunday night. And that flip has consequences: If it is tails, she'll be awake on Tuesday; if it is heads, she will not be awake on Tuesday. Therefore, if she finds herself awake on Tuesday, she knows it was tails. If she finds herself asleep all day Tuesday, it was heads, although she isn't cognizant in this case. Unfortunately, it's not that easy: She doesn't know, when she is awake, if it is Monday or Tuesday. So she must apply Bayesian reasoning. "What would be the probability of my being awake on an unidentified day during the course of this experiment, if the toss was heads? What would it be if the toss was tails?" The number that matters is the ratio of these two values, which is 1/2. This is called the likelihood ratio. In Bayesian reasoning, you multiply the prior odds (1 to 1) by the likelihood ratio (1/2) to get the posterior odds (1 to 2). Odds of 1 to 2 converts to a posterior probability ( SB's *credence*) of 1/3.
She wakes up. She knows she would wake up with heads, and she knows she would wake up with tails, so both are equally possible: prob 1/2. She also knows that if it was heads, then it is Monday. And if it was tails, then it is either Monday or Tuesday, both with prob 1/2. So P(Heads and Monday) = 1/2, P(Tails and Monday) = 1/4, P(Tails and Tuesday) = 1/4. This means P(Heads | Monday ) = 2/3, P(Tails | Monday) = 1/3. At first sight this may be counterintuitive. But still it is right.
@@ronald3836 Why? There is exact math (Bayes's Theorem) that shows she should have a 1/3 credence in heads. She is asked about her credence in a past event, not about the expected behavior of a fair coin. It's not a forecast, it's a forensics problem. So she must take into account all of the clues at her disposal, including the fact that she's awake. She's twice as likely to be awake on a given day of the experiment if the toss was heads.
You misunderstand the Bayesian reasoning. In order for the apriory hypothesis to change, additional information must be provided. In this quasi-problem, no information is provided. She might as well be answering before waking up, as she has the same amount of information then as she does afterwards, that being that she is going to wake up. period. This is a pointless trick question that uses the arbitrarily assigned weight to one of the events making you think it will become more likely. But this is both a mathematical and a logical phalacy. While it is true that there are twice as many opportunities for her to be right by going with tails, each of those only has a 25% chance of occuring for her. However if you view it from another angle, if tails happnes she is guaranteed to get both of those scenarios, wereas heads only gives her the one. So if she wants to be right as many times as possible, she should pick tails, but that is just like saying that because rolling a 6 gives you 1000$ while other numbers get you only one, you are more likely to get a six. But this question is just vague enough to make both answers seem plausible. This is why you dont mix math with philosophy, the correct answer according to probability theory is 1/2.
@@RealAXork *a priori* By definition, an a priori assumption does not change, but let's gloss over that. I don't see how your argument can resolve the Technicolor Beauty problem described by Michael Titelbaum: "Everything is exactly the same as in the original Sleeping Beauty Problem, with one addition. Beauty has a friend on the experimental team, and before she falls asleep Sunday night he agrees to do her a favor. While the other experimenters flip their fateful coin, Beauty's friend will go into another room and roll a fair die....If the die roll comes out odd, Beauty's friend will place a piece of red paper where Beauty is sure to see it when she awakens Monday morning, then replace it Tuesday morning with a blue paper she is sure to see if she awakens on Tuesday. If the die roll comes out even, the process will be the same, but Beauty will see the blue paper on Monday and the red paper if she awakens on Tuesday. "Certain that her friend will carry out these instructions, Beauty falls asleep Sunday night. Some time later she finds herself awake, uncertain whether it is Monday or Tuesday, but staring at a colored piece of paper. What does ideal rationality require at that moment of Beauty's degree of belief that the coin came up heads?" -- Titelbaum, "Quitting Certainties", 2013 She gains no information from the color of the paper as to what day it is, or as to the result of the coin toss, since the results of coin and die are independent. Yet a Bayesian analysis shows that, given she sees a red paper, she must assign a 1/3 credence in heads; likewise, she must assign a 1/3 credence in heads if she sees a blue paper. If she is not allowed to update based on the color of the paper, then she will have the same credence between the moment of awakening and the moment she sees the paper. So she must already have a 1/3 credence in heads as soon as she wakes up, which is analogous to her waking up in the original experiment.
@@rsm3t I am bad with terminology and spelling as you've pointed out, but ignoring that, I don't see how the die introduces anything new to the problem. It is once again a matter of what the question is really asking. If she really wants to know what the outcome of the toss was she should look ak it probabilistically. She knows that the odds of the coin coming up heads are 50% just like the die rolling an odd number. She also knows they are independant events so she is just as likely to find a blue paper as she is to find a red one and this tells her nothing about the outcome of either event. So once again, she gains no information and can't update her hypothesis. But if she wants her answer to be correct more times than not, than she should always bet on tails, since that gives her more oportunities to answer. That is a different question though. But from a philosophical standpoint it's valid to ask which of these two questions she should be asking. If you ask me though, it's kind of a pointless debate.
The question asked to Sleeping Beauty is "What do you believe is the probability the coin landed heads?" The question is not asked to you, an outside observer. The question is not "What is the probability of a fair coin landing heads?" The question is not "Did the coin land heads?" The question is not "If today is Tuesday what is the probability you were asked this question before."
@@tedr.5978 Yeah you mean an explanation where he somehow ends up with more balls than there existed at the start? Among other things, Leftism is an ideology of conjuration magic. The Leftists believe they can conjure stuff out of nothing. Exactly like witches do. And lo and behold, some of the highest ranking leaders of the Leftist cause call themselves, you guessed it, witches! I rest my case.
Sorry for the imperfect English it is my second language but if you have any questions I would love to clarify and if you want to provide counterarguments to any of my claims, I would love that even more. Stay respectful
Good job Marco. There are two outcomes to the coin toss. If the coin comes up Heads then Sleeping Beauty is woken once. If the coin comes up Tails then Sleeping Beauty is woken more than once. The probability of each of these is 1/2. If the outcome is Heads or Tails then SB is woken Monday. The probability of this awakening is 1/2 in each case. However in the case of Tails she is also woken on Tuesday. The probability of the Tuesday awakening by itself is 1. It is certain. It is dependent on the Tails. If the toss comes up Tails then she will be woken Tuesday. Therefore the probability of SB being woken Monday and Tuesday is 1/2 x 1 = 1/2, the same as the Heads outcome. The mistake people make is to treat the Tuesday awakening as having the same probability as the Monday Head or Tail awakenings. It does not. It is a certain result of a Tails outcome with no probability that it not occur. On awakening, SB would realise she does not know when she is being woken or how many times she has been woken previously or will be woken in the future. It may seem that her awakening is one of three possible outcomes, Heads Monday, Tails Monday or Tails Tuesday. Casual observers we might be tempted to split the probabilities equally. However SB might realise that a Tails Monday and Tuesday awakenings are a single result of a Tails outcome in the coin toss with the same probability as a Heads awakening outcome. No matter how many times she is awakened after a Tails outcome the probabilities are the same. That is, more awakenings do not make it more probable that SB is waking up after a Tails. In other words every awakening is the outcome of a single toss of the coin. Hope this helps.
Each wakeup has a 1/2 chance to happen, but as they are equally likel to happen, a 1/3 chance to be the one you are experiencing. So given you wakeup, the chance is 1/3 that you happen to be wokenup while the coin toss was heads.
@@OzoneTheLynx Let's say I flip the coin and let it land on the floor inside a vault where I lock it for 10 years. And then I put her to sleep. The physical event of the coin flip happened 10 years ago. It does NOT matter how many days I arbitrarily choose to wake her up if the coin lands on the tails.
@@OzoneTheLynx Chance is not 1/3 each. You have a 50% chance of experiencing the heads wakeup and a 25% chance of experiencing each of the tails wakeups (as your current wakeup).
*Consider another experiment:* A coin is flipped and if it is Heads you are directed draw a marble from bag A that contains 5 White marbles and 5 Black marbles. If it is Tails you are to draw a marble from bag B that contains 9 Black marbles and 1 White marble. The experiment is run: The coin is flipped and the result of the coin flip is concealed from you. You are presented with a bag and directed to draw a marble from it. You know that the bag presented to you is either Bag A or Bag B but since the result of the coin flip was concealed from you, you are unsure which bag you are drawing from. After running the experiment you ended up with a Black marble. What is the probability that Heads was the result of the initial coin flip? *Discuss.*
@@submanstan7488 I guess this is how a thirder would reason: It is less likely that the coin came up heasds because the black marble is more likely to have come from bag B (which contains more black marbles). Expressed mathematically: P(heads) = 0.5 P(black | heads) = 0.5 P(black | tails) = 0.9 P(black) = 0.5*0.5 + 0.5*0.9 = 0.7 P(heads | black) = P(black | heads)*P(heads)/P(black) = 0.5*0.5/0.7 = 5/14 A halfer would reason as follows: The coin is fair and nothing else matters, thus the probability is 0.5. Expressed mathematically: P(heads | black) = P(heads) = 0.5
Reading it again@@Minkouski I would change my answer. Since the coin is only flipped once and a coin only has two options (excluding landing on its edge) I would now go with 0.5. The whole 5/14 thing is a red herring. We all know that doesn't make any logical sense. Multiple coin flips, however, would change things.
Your explanation here is correct, but for a slightly different reason. The reason why original video is wrong, is because they are answering a different question than they are asking. The question of: "What is the probability that the coin came up heads" has the correct answer 50%. But the question that the original video is actually answering is: "what is the probability that the sleeping beauty is correct when she is woken up and asked "why did we wake you up?" and she states that the reason is that coin came up tails?". The answer to this question is 66.67%, because she knows she is 2x as likely to be woken up from tails then heads. So if she says tails all the time she will be correct 66.67% of the time.
This is the canonical form of the question: "What is your credence now for the proposition that the coin landed heads?" *your credence now* means "your credence, knowing that the experiment is in progress and you have just woken up".
Initially I answered 1/2 however one compelling argument I can see for the thirder position is to consider a slightly different scenario, in which if it's Tuesday and the coin landed on heads then she is still woken up, but within 10 seconds of being woken up she is told that it's Tuesday and the coin landed on heads, while otherwise she is told nothing. In this new scenario when first woken up there's a 1/4 chance that she will be told that it's Tuesday and the coin landed on heads, and once 10 seconds have passed if she has been told nothing then she can eliminate the possibility that it's Tuesday heads and the probability of the remaining possibilities remain equal to each other, and so go to 1/3. Thinking about the original scenario, I would argue that during the times that she is woken up, it's always at 12:00 noon that she's woken up then at 11:59 there's a 1/4 chance of it being Tuesday and that the coin landed on heads, in which case she will not be woken up at 12:00 noon. If it's 12:00 noon and she hasn't been woken up then the probability that it's Tuesday and the coin landed on heads is 100%, however she isn't conscious to consider that it's 12:00 noon and she hasn't been woken up. Still I would argue that she can eliminate the 1/4 chance that it's Tuesday and the coin landed on heads from the possibility space as soon as she wakes up, and so being awake, even without being told information does convey information in the sense that it's not Tuesday Heads and so change the probabilities. When she's asleep there's a slightly higher probability that it's Tuesday and that the coin landed on heads as she could be sleeping when she would otherwise have been awake, although she isn't conscious to consider that.
This clinches the disparity of probability assessment: 16:21 "They are two independent probabilites" The second probability calculation has an added variable. Sleeping Beauty must factor in, not just the coin toss, but also the days such that Tails is weighted by 2 days versus Heads with just 1 day.
It's a question involving the sleeping beauty's first-personal beliefs and the amount of credence she has in assingning probabilitites to the coin flip. The question is not about the probability of the coin turning up heads from a third-person point of view, as you are assuming.
Halvers and thirders answer to 2 different questions. One is "what is the objective probability of getting H or T?", the other is "What is the probability that the sleeping beauty will correctly guess whether it was H or T every time she is put out of sleep?". The sleeping beauty can either give the same answer based on a logical conclusion (skewed probability of guessing H or T) or answer randomly as she can't remember anything. So if we do this experiment for just one week and she answers the question randomly, the chances are on average 50:50. If she sticks to one answer she will either be right 2 times / wrong 1 time or right 1 time / wrong 2 times. If we repeat the experiments for many weeks, from the guessing statistics perspective it's better for her to always bet on T.
Say I draw a card from a standard deck of playing cards. I don't show it to you, but I tell you it is a black card. If I then ask you for the probability that I drew the Ace of Spades, am I asking you for the "objective probability of drawing the Ace of Spades (1/52)" or "The probability of guessing correctly that this card is the Ace of Spades (1/26)?" Neither is correct, although the second is closer. The first is laughably absurd. The question is clearly posed within the state of knowledge that the card is one of 26 black playing cards, not one of 52 playing cards of either color. To make a similar claim about the coin is equivocation, trying to support an argument that is not actually the one used by experts who are Halfers. The second is projecting a philosophy of assigning values to probabilities, called frequentism. I must stress that it is a philosophy, not a mathematical definition. But it does not apply here since the number of trials depends on the result. The actual question is unambiguous: within the state of knowledge that you have at this time, what is the probability of Heads? Halfers argue that this state is no different than the state, on Sunday, that includes another possible awakening. Thirders argue that each potential waking is a different state when viewed from within it.
@@jeffjo8732 , your example is not analogous to the Sleeping Beauty problem. She does not acquire any new information during the experiment and therefore her state of knowledge remains the same during the experiment. However, when you tell me that you drew a black card you give me a lot of new information. Maybe I can clarify this with another example. Suppose that I have two deck of cards, one of which contains only red cards and second contains only black cards. I pick one of the decks, but you don’t know which. I draw a card. What is the probability that I drew the Ace of Spades?
@@user-yx1oj1fs9b The example was to show how to interpret the question, not how to answer it. The point was that "Halvers and thirders answer to 2 different questions," as stated in the comment I replied to, is completely wrong. They are answering the same question, in the same context, but disagree about what that context means. Thirders understand what "new information" is, and know that SB has it. Halfers do not; they never even define it, they simply assert that it doesn't exist. Your "clarification" is completely irrelevant. It is a straight-forward probability problem, not a conditional one. But assuming that the black deck has N cards and M Aces of Spades (you do realize that you didn't specify this, don't you?), the answer is M/2N. Anyway, the prior probability space for any probability experiment includes a sample space; that is, a set of all disjoint outcomes of that experiment. The mistake halfers start with is thinking that the sample space is {H,T}. Since SB's state of knowledge can only ever span one day, the correct sample space is (Mon,H), (Mon,T), (Tue,H), (Tue,T)}. What halfers further don't understand, is that (Tue,H) is an outcome, that SB knows it is an outcome on Sunday Night, whether or not SB understands that she can't actually observe it when it does. But when she is awake, she knows that (Tue,H) is not the outcome she is observing. It can be any of the other three. THIS IS WHAT "NEW INFORMATION" MEANS. She has it, and can use it, and the answer is 1/3. This might be clearer if the testers tell her this: "You will wake up in one, or two, different situations; a fair coin toss will determine which (Heads=1, Tails=2). But we will wipe your memory and put you back to sleep after each, so you will not be able to tell if you have been awakened before. Essentially, each waking will be an independent outcome of this experiment within your state of knowledge. While you are awake...."). There are other ways to prove that 1/3 is correct, but halfers look at them with the same blind eye that they use to claim that (TUE,H) cannot exist.
@@jeffjo8732, thank you for your response. Of course, you are right about my counterexample being underspecified, but it is just fairly simple one and I (rightfully) assumed that you will figure it out easily, which you did. I’ll try to explain what I meant. First of all, I was responding to your cards example which I think was drastically different from what is going on in the SB problem. Suppose I split the regular deck of cards into two packs, one of which contains 26 black cards and another contains 26 red cards. Then I pick one pack without you knowing which one I picked. I decide which pack to pick by flipping a fair coin. Then I drew a card out of this pack and ask you what is the probability that the card is the Ace of Spades. Your answer would obviously be 1/52 (M/2N in your notation for a general case). And this answer is essentially a statement about how the world really works. Funny thing about it is that this answer is sort of absurd since at that point you know for sure that the probability is either 1/26 (if I picked the pack of black cards) or strictly zero (otherwise). Of course you may argue that since you don’t know which pack I have the probability of getting the Ace of Spades is 1/26*1/2+0*1/2=1/52. But if you want to maximize your chances of being right answering the question you have to answer that the probability that I drew the Ace of Spades is zero precisely because you know that I either picked red cards or black cards with the probability of 0.5. Going with zero you have 50 % chance of being right. How so? Well, because you are not trying to make a claim about how the world really works but rather trying to predict as accurate as you can the outcome of this particular attempt given the information you have. And this information is this: you know, that I picked a particular pack of cards. Therefore, there is a 50 % probability that I’ve picked the red pack, and the probability of drawing the Ace of Spades from the pack of red cards is zero. That’s the crucial point: zero is not the wrong answer in and of itself. In fact, it is exactly the right answer for those cases in which I pick the red pack. It is just not right generally. In other words it is the right answer to the different question. The difference if that in one case we are talking about actual probabilities (about how the world really works) and in the other case we are talking about our credence for some particular propositions about how the world really works. If you just draw a card from a regular deck the probability of getting the Ace of Spades is 1/52. What is my credence for a proposition that instead this probability is 0? Well, it is zero, I am 100 % confident that it is not 0. If you follow my procedure and split the deck into two halves and then draw a card from one half of the deck what is the probability of getting the Ace of Spades? Well, it’s still 1/52. But what is my credence for a proposition that instead this probability is 0? Well, 50 %. I am 50 % confident that after you split the deck the actual probability is 0. It is the same with the SB. You yourself stated in the original comment that “The actual question is unambiguous: within the state of knowledge that the SB has at this time, what is the probability of Heads?“. Well, does the state of knowledge that the SB has at any given time include the knowledge that the probability of fair coin landed heads is 1/2? If so (which is the case in the SB problem) why would any additional information alter this? If the SB already knows how the world works (in this case she knows that fair coin lands heads or tails with probabilities of 1/2) she already has exhaustive state of knowledge. However, if we are talking about her credence in some particular outcome of experiment, it is different. She doesn’t know whether the coin landed heads or tails but she knows that she is twice as likely to be asked about it if the coin came up tails. So her credence for that scenario is naturally 2/3, and her credence for the coin came up heads is 1/3. But these are not her estimates of the probabilities, rather her credence for different scenarios of the experiment. In exactly that sense halfers and thirders are answering the different questions. Halfers answer the question “What is the probability of the coin came up heads?” Right answer to this one is ½. Thirders answer the question “What credence the SB should have for a proposition that the coin came up heads?”. Right answer to this one is 1/3.
@@user-yx1oj1fs9b "I was responding to your cards example which I think was drastically different from what is going on in the SB problem." Again, the "cards example" was not meant to say anything about "what is going on in the SB problem." It is meant to show that asking for a probability, after establishing a state of knowledge, is asking for the conditional probability within that state of knowledge. That's all. "Funny thing about it is that this answer is sort of absurd since at that point you know for sure that the probability is either 1/26 (if I picked the pack of black cards) or strictly zero (otherwise)." This is a fallacy. You are saying Pr(Ace of Spades|Black)=1/26 and Pr(Ace of Spades|Red)=0, but ignoring that you are valuating conditional probabilities. So if you consider it seriously, you don't understand probability at all. The exact same logic says that the probability is either 1, or 0. The card either is, or is not, the Ace of Spades. The point of probability is to measure the relative possibilities of a set of outcomes that are all possible. You can narrow the set based on knowledge that the set represents the set. And it works the same way whether that result has not yet been determined, if it has but you don't know anything about it (as you claim describes the SB problem), or you have partial knowledge (as is the actual case). "Does the state of knowledge that the SB has at any given time include the knowledge that the probability of fair coin landed heads is 1/2? " More accurately, it includes the assumption that a fair coin, *_when_* *_it_* *_was_* *_flipped,_* had a 50% chance to land on Heads. But it also includes the knowledge that there were four possible situations that follow from that flip, depending on Heads or Tails, and some are different than what is happening. She has no information, other than the schedule, to determine which of the four represents her current situation. The halfer argument starts with the assertion that is essentially that one Heads situation can never exist. Being unobservable is not the same as having no possibility to exist. It goes on to argue, incorrectly, that the two Tails situation are the same situation, since they are part of the same future on Sunday Night.This is absurd - they are distinct situations, and SB is in only one of them. What if we wake SB on Tuesday, after Heads, but take her shopping instead of interviewing her? The argument that (Mon,T,Interview) is the same outcome as (Tue,T,Interview) also says that (Mon,H,Interview) is the same outcome as (Tue,T,Shopping). Like I said, absurd. Finally, she is not asked for the probability that a fair coin, *_when_* *_flipped,_* lands on Heads. She is asked for the probability, given her current situation that depends, in part, on a (past tense) coin flip, *_landed_* on Heads. These are not the same thing. That's why "additional information alters this." Because the random experiment is not just the coin flip, it is everything that leads up to SB being awake and interviewed. That includes dividing the observation opportunities between Monday and Tuesday.
Veritasium (5:03) "She learns that she's gone from existing in a reality where there are two possible states, the coin came up either heads or tails, to existing in a reality where there are three possible states." Marco (5:24) "This is exactly what I was saying. They are not three states, they are only two states but tails is split up into two sub-states. But these two sub-states are not equivalent to the one heads state or the overall tail state." Both are wrong. Marco's position makes the common error that connects all halfers. The Veristasium position is a misrepresentation of Adam Elga's original thirder argument, which is closer to Marco's but avoids the significance of that error. There are four states (or two sets of two sub-states, which is the exact same thing). The people who want to argue about epistemology (the theory of the scope of knowledge) try to convince us that Tuesday, after Heads, somehow ceases to exist because SB sleeps through it. It can exist, SB knows it can exist unobserved, and SB knows that what she is observing is not that state (or sub-state). So of the four equally-likely states (or two equally-likely states, each with two equally-likely sub-states), SB knows that she is in the only one of three observable states where the coin landed on Heads. And there is an easy way to demonstrate how the "missing" state (or sub-state) must be present in her knowledge. There are actually four different, but equivalent, ways that the experiment can be performed. The day when SB is always wakened can be Monday (as in the video) or Tuesday (by exchanging the procedures on the two days). Then, the option to let her sleep can come after Heads (as in the video) or Tails (and she will be asked about Tails). Each way that it can be performed must have the same answer. So, use four volunteers. Assign a different one of the four possible ways to each, but don't tell them which. Use the same coin flip for all four. On both Monday, and Tuesday, three volunteers will be wakened. Bring them together, and ask them, as a group, to assign a probability to each individual, to the proposition that this is her only waking. Here, each of the volunteers can be considered to represent one combination of (Coin, Day). Three "exist" in the sense that they are in the awake group, and the fourth "exists" in the sense that they know about her, and know that she is asleep. They also know that exactly one, of the three who are awake, will only be wakened once. And that there is nothing that allows them to say that the probability of being that one is different for any of them. That probability is 1/3.
I explained to a dozen 1/3 ers why the answer is blatently obvious to be 50/50 snd no amount of logic or reasoning changed their mind. It is scary to me that these people might be on a Jury trial, or otherwise affect someones life.
No, what is really scary is that you think the calender skips from Monday to Wednesday if we don't wake Sleeping Beauty. There are four possible states where we have to decide whether to wake SB. One is _eliminated_ as a _possibility,_ not _removed_ from _existence,_ based on the evidence that SB is awake. When outcomes are eliminated based on evidence, but still exist in the experiment, Mathmatics tells us to update the probability to: P(H|Awake) = P(H&Mon)/[P(H&Mon)+P(T&Mon)+P(T&Tue)] = 1/3.
@@andyh8239 You're joking, right? When SB imagines, on Sunday, what Tuesday will be like, is it: 1) There are four distinguishable (not to me, to the experiment), equiprobable states of the experiment that I could exist in over the next two days. It can be Monday after Heads, Tuesday after Heads, Monday after Tails, or Tuesday after Tails. I won't be aware of Tuesday after Heads, BUT THE EXPERIMENT STATE EXISTS REGARDLESS OF MY WAKEFULNESS. 2) The states Monday after Tails and Tuesday after Tails will exist at the same time, since my conscious memory will experience both but not be able to distinguish them. The state Tuesday after Heads cannot exist this way, since it is distinguishable from Monday after Heads be if it does exist. The answer to the question is (# observable states with Heads)/(# observable states). That's 1/3 in interpretation 1, and 1/2 in interpretation 2. In fact, the only way to get 1/2 is interpretation 2. Most Halfers understand the "Monday same as Tuesday" part, yet somehow accept it. They don't realise that the "Tuesday ceases to exist" case is required as well. If you doubt me, change the problem so you wake her on both days, but don't ask for a probability on Tuesday after Heads. There are other, more formal ways to prove the answer is 1/3. Halfers ignore them like Trumpers ignore stolen documents.
But it changes the probability of guessing the result of the coin flip correctly. In this case there’s just a probability of guessing twice. But even just a probability of guessing twice will change the probability of guessing correctly.
@@laowei7279 No. She's not being asked that. She was asked what the odds of a 50/50 probability were. It doesn't matter what day it is, what actually happened, or anything else. The odds were 50/50 period. Even if it IS Tuesday the probability didn't change.
My take: Given that I sleeping Beauty am awake, and given that if the coin lands tails I will wake up a million times, it is more likely that tails has come up, but if the question is, given that I am awake, which SET (the set of one wake versus the set of plural wakes) am I in, then it’s equally likely in each every instance of waking that the coin was either heads or tails. Subtlety different questions. Am I wrong?
Well that set should have an equal probability to the sum of its compoments since they are disjoint events. So if you think the set has a total probability of 1/2 it makes no sense for those million wake ups to add up to any more or any less. What I think confuses people is the fact that you are guaranteed to wake up all of those times which makes them feel like waking up is more likely, but what you forget is that due to the memory wipe, each of those wake ups is entirely dependant on the others and only one can occur from beauties perspective (she forgets the others). This means that the probability of her waking up for one of those days is 1/2n where n is the number of days she is going to wake up for tails, and 1/2 for waking up once for heads.
Mathematically, there is no paradox. The two answers are correct depending on how the question is interpreted. However, I believe that there is something hidden here. Mathematics is an abstract view of reality. Reality is much more complex. I do believe that there is an opportunity to introduce a new concept that clearly defines the difference. A new mathematical symbol or equation that describes the change. This could have a true impact of our understanding of reality.
You are correct in one way. If interpreted "when it was flipped, what was the probability it came up Heads?" then the answer is 1/2. And if interpreted "based on your state of knowledge now, what is the probability it came up Heads?" then the answer is 1/3. But you are wrong that the first interpretation is a valid one. Say I ask you to draw a card, but only let me look at it. If I tell you that it is an Ace, is the probability that you drew the Ace of Spades 1/52, or 1/4? Both questions clearly want you to use the information you have. True Halfers think that information does not allow you to say the answer changes between the two interpretations.
Not trying to hate but this video (and a lot of the comments) seem a bit nitpicky to me regarding what question Derek was asking. Your tone and the comments strike me as a little condescending which is a shame. I would have loved to watch a video that makes good/ logical points and maybe even gives a definitive answer to such a controversial problem. But since your other viewers and commenters don't indicate to feel similarly you are well within your rights to ignore this comment I guess. (: But if you'd like some constructive criticism in case you want to grow your community and not have first time viewers like me leave when they click on your videos: Maybe try to use more auxiliary verbs and even if you're fairly sure that your conclusion is right, treat it as an optionon. That makes it easier to stomach for people that might disagree with you thus they won't click away immediately. Also, I thought it came across as a little mean (although you probably intended it to be funny) when you edited 1:56-2:18 and asked sarcastically "so what is the problem here". If the point of your video was to argue that it doesn't matter when you ask an individual in the 'sleeping beauty' role what the probability is then this video could have been a lot shorter. Instead, you picked apart evrything Veritasiums' video argued with the same argument which unfortunately came across as very spiteful. I hope you don't see this as a personal attack and whether or not you choose to agree with me or not, I hope you have a good time on youtube.
It was just too smarmy for my taste, you can give your response without the "looping the video with slow-mo" bit and using a condescending tone - what an unlikable response
This is utterly frustrating. Stop trying to rephrase the question. We are asking Sleeping Beauty if she thinks the likelihood she's been awoken by a heads or tails flip. Maybe he should take his own suggestion and rewind, and listen again. It IS NOT asking the likelihood of a coin landing heads or tails. Now, with fresh perspective, if you're woken up twice each time the flip is tails, can you not see the other argument that from her pov, heads has only woken her up 1/3 the time?
Ok then riddle me this: If the coin lands heads, she is neither woken up on Monday or Tuesday, but if it lands tails, she wakes up on Monday and Tuesday. Would you still say that she should say "there is a 50% chance the coin landed on heads" in this case? After all, that is the probability of the coin.
In this case waking up gives you information, namely that you woke up. The conditional probability of tails given that you wake up is 1. In the original problem, waking up does not give you any new information. But you are right that this is important. Puzzle: in the original problem you wake up and you are told it is Monday. What are now the probabilities?
@@ronald3836 "Puzzle: in the original problem you wake up and you are told it is Monday. What are now the probabilities?" also 50% each. David Lewis is a fake halfer.
@@Feds_the_Freds I realised my modification does not result in a well defined question. By telling SB the day, you change the rules of the experiment. If she understands that the rule is now that she is always told the day, then the probability of heads when she is told it is Monday is 1/2. But if she assumes she is not supposed to know the day but by a fluke event gets to know that it is Monday when she wakes up, then she should say 2/3.
@@ronald3836 what do you mean by it being a fluke? Something like: She isn't supposed to know the day, but finds it out anyways? how would you well define that? Maybe something like: the researchers toss a parallel coin and tell sb the day, if that parallel coin lands heads. So P(Heads|Heads AND Monday) = 25% P(Tails|Heads AND Monday) = 25% hm, still 50/50. I agree that she should say that it's 2/3 heads given that she finds out the day (Monday) by a before unknown event but how to construct the experiment? Do we tell sb that it's the normal sb problem but what we actually do is "spoonfeed" her the information on monday anyways, which day it is? Like, a reasearch assistant goes in and says "have a good monday evening" and the interviewers are seemingly angry that he gave sb additional information that she wasn't supposed to know? Is that modification to the experiment actually what David Lewis meant?
@@Feds_the_Freds yes, that is what I mean. I'm not sure how to define it properly. But you could think of a scenario where unexpectedly a siren goes of on the first Monday of the month (in my country sirens used to be tested on the first Monday of the month), which tells SB it is Monday (or war broke out). SB will understand that this was not an intended part of the experiment. I think now she should estimate the probability of heads to be 2/3. Before she heard the siren, it was 1/2 for H on Mon, 1/4 for T on Mon, 1/4 for T on Tue. After the siren she knows it is not T on Tue, so the other two become 2/3 and 1/3. But this is really subtle and perhaps I am wrong. If instead of the siren there is an envelop that she can open to learn the day (and she knew this envelop would be there), and she has to make guesses before and after opening the envelop, then I think she should say 1/2 before opening and again 1/2 after seeing it is Monday.
From her perspective, it doesn't matter the actual coin probabilities, but the probability of being awakened with a certain result in the last coin toss. If the experiment is repeated many times, she will be awakened following heads twice as much as following tails. If she receives a dollar every time she guesses the result correctly, she should answer heads every time.
Where do thirders get it wrong? In this forum and others many people prove in different ways that 50% is the good answer, but now here is my take on exactly where the thirders get trapped in their reasoning (not speaking about those who answer the wrong question) The confusion arise if we say that in Sleeping Beauty’s universe there is no Tuesday Heads. That is the false assumption. She lives in the same universe of all the people around her, with the only difference that in this situation she is sleeping and asked no question. Let me rephrase the problem in a way that it is easier not to get trapped in the reasoning that there is only 3 situations. Don’t tell SB that she will not be awaken on Tuesday Heads. Before being put to sleep she knows she has 25% chance to be in any one of the 4 states when she will be waken up. Now when she wakes up ask her the confidence of the coin flipped Heads. 50% she will answer, no debate here! Now tell her we are not in the Tuesday Heads state. This is a new info but that does not change the probability the coin went Heads. The only new info for SB is: « if the coin actually went Heads, then I know for sure the day is Monday. So my confidence of Monday Heads rise from 25% to 50%, my confidence for Monday Tales is still 25% and also Tuesday Tails 25%. » The new info of not being in the Tuesday Heads state does not change the confidence the coin went Heads or Tales, it rises her confidence of being Monday Heads. With the original problem, telling in advance that she will not be waken up Tuesday Heads is just the « new » info given in advance « hey SB, today is not Tuesday Heads »
You are absolutely right that the big mistake is thinking that there is no Tue+H. There is. But then you go off into a wild rationalization for why the incorrect answer should be considered correct. There are four possible states of the experiment. {Mon+H,Mon+T,Tue+H,Tue+T}. Each has the same probability to be the state of the experiment at a random point during it. But when SB finds herself awake, it is an observation that the state is not Tue+H. It doesn't matter what would happen in the state Tue+H, as long as it is different than "Ask her for the probability of Heads." That includes not allowing her to observe it, but you could just as well wake her and take her shopping instead of questioning her. It is trivial to conclude that the conditional probability of Heads, given the observation that the question was asked, is 1/3. But there is another mistake you make, when you say "This is a new info but that does not *_change_* the probability....". This is a common mistake made by beginners. A conditional probability is a different measure than the unconditional probability. It is "filtered" by the information you have that eliminates some states. So it always is a "change." Sometimes it may change to the same value, but the calculation that determines the value is different. And that is the reason halfers make their big mistake. They do not see that something has been removed from the set of possible observations, because they concentrate too much on that fact that SB knows an observation will be made.
The probability of Heads is not conditional to her observation of being awake. Giving her the info of not being in the tuesday-heads state is not a new info about the outcome of the coin. She already knew at 100% she wound wakeup on any on the 2 outcomes
@@jonathanlavoie3115 How is it not new information? Please, provide a definition of "new information" that supports this assertion. And no, it does not have anything with knowing she would be awake. It has to do with what she knows about the experiment at the moment. Try it this way: Say that the coin you flip on Sunday has a value of 1="Heads" and 0="Tails." Get a second coin, a different kind of coin, with a value of 2="Heads" and 0="Tails." On Monday morning, place this second coin showing Heads. On Tuesday Morning, turn of over. The value of the experiment at any point in time is the sum of these two coin's values. Now, wake her on both days. For Question #1, ask her to write down a sample space for the current value of the experiment, and to assign a probability to each possibility. Then, administer the amnesia drug and, if it is Tuesday or the first coin is Heads, the sleep drug. Allow her to see her first response, and ask Question #2: How would she update it? After she does (or does answers, administer both drugs. The correct sample space for Q1, on either day, is {0,1,2,3}. Each of these is definitely a possibility. The correct probabilities are each 1/4, because no probability can be different than any other and there are four of them. She could even predict this on Sunday Night, raising the question of whether Q1 is needed at all. For Q2, she knows that the value can't be 3. But the sample space she arrived at yin Q1 (or Sunday Night) still applies, as a _prior_ sample space. The fact that 3 is no longer possible is a classic example of "new information" because it eliminates a possibility from the _prior_ sample space. It can be used to update the probabilities of the remaining values to 1/3 each. Now, how is this different? Does the information she has in Q2 of this experiment differ, in any way, from the information she has in the usual version? Doe the _prior_ sample space not apply if the same two coins are used? The halfer's error is that they consider value=0 and value=2 to represent the same result of the experiment. It does for the people running it, at different times, but SB only knows of only one instance.
« Allow her to see her first response, and ask Question #2 » This is new information, you’re basically telling her we are Tiesday. In SB experiment, she has NO clue if this is Monday, or this is Tuesday and being ask the same question for the second time.
@@jonathanlavoie3115 No, in the we are actually telling her that it is not Tue+H, the same information she gets in the common variation (which, actually, is not the original problem). She still knows that Mon+H, Mon+T, and Tue+T are possibilities. So, what is your definition of "new information" that allows an update? What makes you say this doesn't exist? Mine is that, as an answer to the first question, she says the sample space is {Mon+H, Mon+T, Tue+H, Tue+T} and that each has a *_non-zero_* probability. But after amnesia, Tue+H has *_zero_* probability. The sample space changed. This is "new information." Or did you ignore me asking for a definition because you don't have one?
First of all, really nice vid with your explanation and sharing your thought processing! Thank you for that and very much appreciated! Furthermore, I do agree on you regarding the possible states. There are only two states [Monday heads] OR [Monday tails AND Thuesday tails]. In the end its still only two states in total (that's also how you logical built up, stuff as a programmer). Moreover, the question and given information is incomplete if you ask me. In fact the information given by Veritasium contains already a small difference vs the explaination on Wiki. If you truely want to answer this question from the perspective of Sleeping Beauty (and not as an outsider perspective), you need to exactly know, in detail what information is kept (eg. procedure details and if this information is lost to her during the experiment). Movreover, you also need to know, if information can be collected during a wake-up-moment. Without knowing what type of information Sleepy Beauty is holding during the expirment, you cannot answer this question from her perspective (as it was ment to). This information is important because sometimes you can not proof something but can rule out all the other things, to still come to the same conclusion. Also the question can be answered from multiple disciplines, one from a purely math point of view, but also from a psychological point of view. To give an example; Let assum she is aware of the expirment. In otherword, she exactly knows the procedure and all the rules of the expirement (possible states) during the expirment. In that case you has the possibilty to back-track days. However, assuming she doesn't know she is part of an experiment, than you can also argue she isn't even aware about the two sub-states at all (Monday tails and Thuesday tails). From that point she only get the information from this single question -> "What is the probability that the coin came up head?". So based on this information the can conclude that a coin has been flipped. And based on this information only, she will definitly answer the 50/50 change (excluding the minuscule change that the coin came up on its side 😅). Anyway keep up the good work and have a blessed day!
Even from the perspective of Sleeping Beauty the probability would be 1/2 heads and 1/2 tails. Remember, she does not remember how many times she awoke so from perspective it's always the first time, so it is always Monday. But which Monday? Monday of heads, or Monday of tails? The two Mondays are NOT the same Monday.
1. Assign an outcome on the result of a flipped coin. (Heads will have SB wake up once on Monday. Tails will have SB wake up twice on both Monday AND Tuesday.) 2. Define the coin as being fair, meaning it has a 50% chance to land heads and 50% chance to land tails. (This is the answer to the question, "What is the probability SB woke up because of a heads coin flip?") 3. Execute the pre-decided outcome for the coin flip. (This is Derek's video. The coin lands on tails, therefore, SB will be awoken twice on both Monday AND Tuesday, and each time she is awoken she will be asked the question "What is the probability that you woke up because of a heads coin flip?" to which her answer will be, assuming that she is intelligent enough to understand how probability works, "It is 50%." Considering that the coin landed tails she will be asked two times the same question. And both of those times, assuming she is intelligent enough to understand how probability works, she will answer the same way. Because why would she decide to answer any different way when she knows the rules of the game before she starts playing it? Why would she answer any other way when she knows the coin is fair? And why would it matter if she gives the correct answer or the wrong answer? Her answer, correct or wrong, will NOT affect the probability that she awoke as a result of either heads or tails anyway. Because it is the the coin that which determines the probability, NOT her opinion of it.)
Actually, in the original problem, the question asked to her is “What is your credence now for the proposition that the coin landed heads?" It ends up being a problem of how you understand the question.
But the person who originally posed the problem then went on to "prove" using conditional probabilities that the correct answer is 1/3. So this was meant to be about probabilities. (He makes a subtle mistake, though.)
@@ronald3836 Veritasium went on to prove both answers are correct answers based on how you understand the question. If you want to be correct, it’s 1/3 and if you want the probability % it’s 1/2. The question isn’t meant to be about anything rather than being paradoxical and the answer is that it depends on how you understand the question. This whole problem proves that credence isn’t strictly based on objective probability.
Ok, but what is "credence" then? If it is exactly defined mathematically via probabilities then Marco's probability analysis is the right way. If "credence" is just a subjective feeling that cannot be evaluated as "correct" or "wrong" then any answer is OK as long as it's the actual subjective feeling of the sleeper
@@charonme Adam Elga's paper says "credence" but then treats it as a probability. He "proves" that the answer is 1/3 using conditional probabilities (but makes a questionable/incorrect assumption when doing so).
@@charonme credence is how much someone believes in a statement someone else made. If the statement is made by oneself, one’s belief in his own statement is called confidence.
This is a fake paradox abd great litmus test to figure out who actually thinks and who thinks they think. 50/50 is 50/50 regardless of what day it is. SHOCKER. 😂
Suppose you are an extra participant who is also getting woken up and going back to sleep without memory. For you though, you’ll be woken up on Monday+Tuesday for both heads and tails. You’ll also be woken up an hour earlier than SB, and get to observe whether she wakes up. When you first wake up, I’ll assume you’d say P(heads)=50%. Does your probability change an hour later? Certainly if SB doesn’t wake up then P(Heads)=100%, because that’s the only possibility. In the case SB does wake up, can you update your probability?
Indeed. Bayes' theorem provides the formula for updating your credence that the coin landed heads conditioned on the information that Sleeping Beauty was seen to be awakened. The result is P(H) = 1/3.
"if you can't go out on surface, you have to go deeper". Let's start from verifying if what SB lives doesn't influence the answer; imagine the experiment is this: "if it's head, you'll never be awakened, if it's tails, you'll be awakened on monday and then on tuesday. When you'll be awakened you won't remember if you've ever been awakened previously and you''ll answer the question: "given you are awaken (and remember the rules of the experiment), what is the probability the coin flipped on heads?". In this scenario the SB will be awakened ONLY after a tails, so how can she think there was a probability that the coin flipped on heads if in that case she wouldn't be awaken? This shows that the question isn't "what is the probability of a fair coin to flip on heads?" and that the question "what result do you think awaken you today?" (which is equal to the original question) is different, because in this question the event "Awakening" comes to play... It's true that the fair coin gives 2 results, so 2 different events, but is also true that the question doesn't focus on those 2 events (which are 2) but on these 3 events (which are 3, not 2): - today is monday AND the coin flipped on heads; - today is monday AND the coin flipped on tails; - today si tuesday AND the coin flipped on tails. All these events (these "Awakenings") have the same probability to happen BUT only 1 is consequence of the flip of the coin on heads, so the right answer to THIS question ("given you are awake...?") is 1/3. P.S. mathematically it can be demonstrated easily, if the steps are done coherently with the laws of the probabilistic math. Also, at 7.12 the split of 50% of probability is wrong; both the events have 50%, because the second happens the 100% of the time when the first happen, which has a 50% of probability to happen, so both share the same "global" probability to happen. P.P.S. the probability to be AWAKEN on monday after a flip on HEADS ISN'T the probability of a fair coin to flip on HEADS. P.P.P.S. initially i thought i could simplify the problem focusing only on the probability of the fair coin to flip on Heads (50%) but i understood this was a wrong passage after a little thinking on it.
@@hypersonicpiano6120 2/3 of the wakeups result from a tails coin toss, however you are confusing 66% of the total wakeups with a 66% probability of a tails coin toss! These are two very different things.
@@hypersonicpiano6120 The number of times you wake up is irrelevant to the probability of the coin toss because the coin is tossed ONLY ONCE and that is BEFORE you go to sleep. Even if you were to wake up a million times in a row you would still exist in the second of the two possible outcomes. There can ONLY BE two possible outcomes. One outcome for heads, the other for tails. And since the coin is a fair one, meaning it's a magical coin that will randomly generate a result at 50% chance then you already know the probability that you awoke because of a heads. Heads (50% chance) {State ---> Wakes only once on Monday} Tails (50% chance) {State ---> Wakes twice on both Monday and Tuesday} Heads or Tails are the ONLY states. In the parenthesis is the probability of the coin landing in that state. In the squiggly brackets is the action/s taken within that state. In computer programming you would write it thus. If (H == TRUE) then (A == TRUE) Else If (T == TRUE) then (B == TRUE AND C ==TRUE) The Monday of Heads and the Monday of Tails are NOT the same Monday. They exist in different states. With an additional difference being that the Monday of Tails is ALWAYS followed by a Tuesday of Tails. Because the Tuesday of Tails exists in the same state as the Monday of Tails.
@@hypersonicpiano6120 The question is what is the probability that the coin landed on heads. In both heads and tails she will awaken on a Monday. But only in the tails will she be awoken a second time on Tuesday. There are only two states she can be in based on the conditional statement of the coin toss. The reason there can only be two states is because the random generator we are using ONLY has two possible answers, heads or tails. If we wanted six states then we would use a six sided die. But here we use a coin. When the coin lands on heads she is awoken one time. When it lands on tails she is awoken two times. The number of times she is awoken does NOT influence the probability of the result of the coin toss. Especially considering it is a fair coin whose probability is already defined before the toss is made as being 50% for heads and 50% for tails.
@@hypersonicpiano6120 if you are right then you would observe that an awakening on monday would happen only with the 75% (50+25) of the probability, since the awakening on monday has 100% of probability to happen (in both cases, with Heads or with Tails, the awakening on monday will happen) your assumption is wrong.
Each wakeup has a 1/2 chance to happen, but as they are equally likel to happen, a 1/3 chance to be the one you are experiencing. So given you wakeup, the chance is 1/3 that you happen to be wokenup while the coin toss was heads.
"Each wakeup has a 1/2 chance to happen, but as they are equally likel to happen..." You just contradicted yourself. Each wakeup cannot have both a 1/2 and 1/3 chance. In fact they can only have a 1/2 chance. Because the wakeup of tails in Tuesday cannot exist with the tails of Monday. Therefore, all the awakenings in tails are equivalent to same exact state of tails.
@@kaizokujimbei143 No I didn't contradict myself. Let me elebaorate on the 2 likelihoods involved in my comment: Likelhood1: For any of the wakeups the likelihood that said wakeup will eventually happen, is 50% (because it is only determined by the fair coin toss). (Hence why the tally in the og video shows the same number for each wakeup). Likelhood2: Because they are equally likely to happen eventually the chance your current wakeup to be any specific wakeup is also equal, because their is no way of differentiating between them and as their are 3 options said likelyhood is 1/3 (i.e 1/3 chance that your wakeup is the heads one). Or to put it in simpler words likelyhood 1 is "the chance that X would happen", while likelyhood 2 is "the chance X had happened (given information y)" The latter case involves a gain in information because of knowing to have woken up. This is made obvious if you replace each wakeup with a chance to be woken up. Example: If heads-> a 1 time 50% chance of waking up If tails-> 2 times a 50% chance of waking up. If you are woken in this scenario it's suddenly pretty obvious, that you gained some information about what happened, because the chances to be woken even just once is significantly higher if it was tails.
@@OzoneTheLynx The question is NOT discussing whether each wakeup will happen eventually. The question is what is the probability that any one wakeup is the result of a heads. And the two wakeups that happen in the tails are linked to one another. They cannot happen independently from each other. Therefore, their probability is the exact same because they both happen under tails. They happen sequentially after tails. If the tails Tuesday happened then you know that a tails Monday happened. Therefore, there are ONLY two possible states. One state for heads Monday, and one state for tails Monday and Tuesday. The reason there are ONLY two states is because the random generator we are using is a coin which can either take values of either heads or tails. If we wanted to have six states then we would use a six sided die. The number of times you are awaken in a single state is irrelevant to the probability of the coin toss. The coin toss and whether it landed on heads is the question. To help you understand even better. Let's say in the heads you are awoken 5 times and in the tails you are awoken six times. The probability of the coin landing on heads is still 50%. Because it is a fair coin. It will land on heads 50% of the time. And it will land on tails 50% of the times. The number of times you are awoken in each scenario does NOT influence the probability of the coin toss.
@@OzoneTheLynx Why is this so difficult for you to understand? Derek literally gave us the answer at the beginning. There is no mystery here. At least in regards to this thought experiment about probability. No. The only mystery is WHY do people keep insisting on constructing false narratives around imagined information when the answer is already staring them in the face? Why are you doing this? You are not smarter than the obvious. You are not smarter than common sense. This is simple logic. It is the same logic used in computer programming. Am I to assume you are not a programmer or that you are so stupid that you couldn't never learn how to code even if your life depended on it? I mean, dude, come on. Use your head already. Derek himself is so obviously trying to confuse you by giving you irrelevant information or by jumping from one point to another making you erroneously think that he is convincing you of something.... magical. There is no magic here. Please, focus.
@@OzoneTheLynx And I would further go on to say that Derek's video is a social experiment. Did you notice that the title of the video updates the like to dislike ratio? I think Derek is trying to prove that the opinion with the most number of people always gets bigger and bigger. He also proves that most people are easily manipulable, hence why they believe illogical and non-factual information. You are not operating under the guise of logic and understanding. You are operating under the belief that you can somehow crack the code of the matrix. Hence the reason why Derek claimed that is people like you who believe in the nonsense about a simulated universe. We do not live in a simulation. So stop trying to pretend to be smart. And for your information, I stopped coding computers about 7 years ago. And I never went very far to be honest. My knowledge is small. And yet I understand a basic conditional statement when I see one. You can learn too. But you have to stop whining like a child and grow up. You cannot break logic. You cannot break objective reality. Period. End of story.
I think this simplifies it: Once she wakes up, she is asked what was the result of the flip. If she answers wrong she is killed. What is the probability she lives? 1/2. Now if instead of that, she only gets slapped in the face for each wrong guess (after the experiment is over), what is the probability she gets slapped? 1/2, but she gets slapped twice if she guesses heads and its tails. Conclusion: she should bet on tails, knowing she has a 50/50 chance of getting slapped and a 100% chance of getting slapped twice as many times if she goes with heads. The phalacy here is to treat the two days provided by tails as independant, as they are bound to happen either both or not at all. But, if you change it up again and say that she gets slapped momentarily and forgets that too, then she only gets slapped once for both bad guesses. And I think this is the most realistic interpretation of the original question, since it implies she won't remember anything that happened during the experiment and thus, her experience should be viewed seperately for each day and not accumulatively. So no, 1/3 doesn't make sense in any scenario that doesn't assume she will somehow count her strikes at the end of the experiment. It makes no difference to her how many times she gets it right otherwise. And if we're talking parallel worlds? I'm pretty sure those don't add up. I may have infinite lives but I can live only one.
I was a halfer but on further reflection I became a thirder. Here's a simple way I explain it: Imagine the experiment is carried out multiple times. On average she will be woken 1.5 times. For simplicity, let's say we run the experiment twice, therefore she is woken an average of 3 times. On one wakening, the coin was heads; on the other two it was tails. The probability it was heads, therefore, is one in three.
Yes, the probability that she will answer the question correctly if she says "heads" is 1/3, because she is answering the question some redundant amount of times, according to the problem. But the question isn't asking her to be correct about whether it came up heads or tails the last time it was flipped. It's asking her what she thinks the probability is that it came up heads. If the coin is flipped tails, yes she will be answering the question twice, but that's irrelevant. It was still a single coin flip. Don't count the number of times she answers the question, just count the coin flips. That's what demonstrates the actual results. The question's a bit vague "what should she say...?" and it leaves some room open to interpretation. But for me, I think she should give an accurate answer, not one that protects her from being wrong more often. If the question were more specific like, "What should she answer, if she wants the record of her answers to match probabilistically to the flip result for each time she was questioned", then I would agree should should say heads is 1 in 3.
I think "being right" and "correctly identifying the probability" are really the same thing. Though I understand your objection. And now that you mention it, since the coin is only flipped once then...1/3 doesn't make sense. Interesting! :) Maybe I'll go with: if the experiment is repeated multiple times then it's 1/3; but if only performed once then it's 1/2.
None. The simulation argument doesn't predict our existence or nonexistence (which would correspond to SB being awake or asleep). It predicts we would observe a Universe that has limited resolution -- something we have not observed.
@@rsm3t What about quanta? What about the limited speed of light? What about uncertainty? Those fit the idea of limited resolution. And btw: the simulation could also dictate that we can't observe resolution limits.
@@wassollderscheiss33Michael Titelbaum analyzes quantum cases in his book, "Quitting Certainties", if you are interested in that. It's too lengthy to go into here. The finite speed of light has no bearing on the problem.
This is what happens when who doesn't have the instruments to make logical thinking expresses an opinion... A great lack in comprehension of conditional probability in all the replies. Anyway here's a hint, the big part of the mathematicians or don't give a f about this problem, or argue on what is the real question...
Everytime I get convinced by a halfer argument I go back to this empirical example at 7:36 and cannot reconcile the two. If I know that "I am currently woken up" and the cases of me being woken up are counted on this paper, and there are 2x many cases of me being woken up that belong to the "tails" column, then it is logical to me that my current awakening is more likely to belong under "tails" column, and so... more likely that the coin came up tails? I don't know where is the error in this line of reasoning
The error is that the number of wakeups are completely irrelevant to the probability. Consider this: We flip a coin repeatedly. When heads comes up I give you $1. When tails comes up I give you $2. In the end two thirds of the money you get will come from the tails results. Does that mean that tails was more likely to be thrown? No, it just shows that the rewards were different for heads and tails. So by saying the number of wakeups equals the probability of the coin you are comparing two things that are fundamentally different (an absolute value and a percentage).
@@32fw34dgasdfg I don't know, he tosses a fair coin and just fills the columns accordingly... so what's the difference between choosing one line from this table at random (probability 2/3 of choosing the one with tails) vs waking up and so also being in a state corresponding to one of these lines (not 2/3 anymore, but why)
@@kskcp7243 Why is it so difficult? The number of times you wake up is irrelevant to the probability of the coin toss because the coin is tossed ONLY ONCE and that is BEFORE you go to sleep. Even if you were to wake up a million times in a row you would still exist in the second of the two possible outcomes. There can ONLY BE two possible outcomes. One outcome for heads, the other for tails. And since the coin is a fair one, meaning it's a magical coin that will randomly generate a result at 50% chance then you already know the probability that you awoke because of a heads. Heads (50% chance) {State ---> Wakes only once on Monday} Tails (50% chance) {State ---> Wakes twice on both Monday and Tuesday} Heads or Tails are the ONLY states. In the parenthesis is the probability of the coin landing in that state. In the squiggly brackets is the action/s taken within that state. In computer programming you would write it thus. If (H == TRUE) then (A == TRUE) Else If (T == TRUE) then (B == TRUE AND C ==TRUE) The Monday of Heads and the Monday of Tails are NOT the same Monday. They exist in different states. With an additional difference being that the Monday of Tails is ALWAYS followed by a Tuesday of Tails. Because the Tuesday of Tails exists in the same state as the Monday of Tails.
@@kskcp7243 2/3 of the wakeups result from a tails coin toss, however you (and Veritassium?) are confusing 66% of the total possible wakeup states with a 66% probability! Both are percentages, but these represent two very different things as per Marco's response. The betting odds of a sporting event do not affect the chance of a particular outcome even though most of the time the betting companies are correct (correlation vs causation).
You misunderstand that what Veratasium is making isn't a mathematical argument but a philosophical one. The math is there just to confuse people. And philosophy is there, as always to make pointless assertions that sound clever.
To be specific, he views the wake up states as aposteriori "realities" in which the beauty finds herslef in and asks how one should look at these realities. The one third argumant is that because in the case of tails you have 2 realities in which you are correct, you should go with tails, while the one half argument tells you that you don't actually know what the right answer is and are just as likely to be wrong or right, regardless of how many times that may be. Indeed the right answer depends on the question which was cleverly presented as "what should she say?" She can recite the bible for all I care, I don't know the woman!
Great video, Derek was not logical in his initial video and misrepresented the "halfer" position, while contradicting mathematics itself to advocate for the "thirder" position. If he was unbiased and logical as you were then the answer would have been obvious from the start. The crux of this problem is that expected value (1.5 awakenings) has been incorrectly conflated with a 1/3 probability.
Sure, so why the Sleeping Beauty has to live 3 events, which are the 3 possibile awakenings, when the coin has only 2 possibile consequences? And if the SB wouldn't be awaken if the coin flips on Heads and would be awaken only after a Tails how could she answer 1/2 at the question (and read well what is the question) "given you are awake, what is the probability the coin flipped on Heads?" That's not "what is the probability of a coin to flip on Heads?" since this question doesn't include in itself the events "awakenings".
@@antog9770 all of these questions are answered in the video, please rewatch it. These 3 live events are really two events. The fact that she is awoken twice when the coin lands tails does not change the 50% chance that it will land heads, she wakes up once, and that is the end of it. 50% of the time the coin will land heads, and in 100% of those cases she wakes up on Monday (50% total probability of taking place and that being the case when SB wakes up). 50% of the time the coin will land tails, and in 100% of those cases she will be awoken on Monday, and in 100% of those cases she will also be awoken on Tuesday. Therefore when SB opens her eyes if you tell her it landed tails there is 50% chance it is Monday and 50% it is Tuesday (because both events are equally likely - guaranteed - to happen). Multiplying this by the overall chance of a tails coin flip (50%) you get 25% chance the current awakening is Monday T and 25% Tuesday T. Therefore when SB wakes up the probability distribution that her current awakening is one of the scenarios is 50/25/25, not 33/33/33. Since both tails awakenings take place, Veritassium counts both in the table which is an incorrect way of simulatung this situation. If you simulate it correctly you will get the 50/25/25 distribution (don't just count the total number of awakenings, but the chance that the current awakening is one of the 3 scenarios).
@@hypersonicpiano6120 there's too lack of probability knowledge to continue, you should study a bit of probability first to think to have understood this problem...
@@hypersonicpiano6120 i said were you fall, MH is consequence of the flip, MT is consequence of the flip and so MH+MT=1 so the probability laws are all happy, then, only when MT happens (which isn't the flip of the coin event, this is another event) surely happens TT, so it has the same probability of MH to happen, and since MT =1/2, TT =1/2 too. The sum is greater than 100% because the TT event isn't a direct consequence of the flip (there will never be a TT first of the MT event), and when you include events which are not consequences of the causes you are considering the sum can't be 100%. Differently by the coin, the SB is directly connected to all of these 3 events, in facts she can answer in 3 possibile scenarios and in 2 of these the coin flipped on Tails, so, since she has to answer the question considering all the info she can have, the right answer is 1/3 since only in 1/3 of the events she can live the coin flipped on Heads. If you understand well the question you can understand all that comes after...
In philosophy people can talk endlessly. In math you can do the work to get to the one and only correct answer. In math the definition of probability is "the number of times a particular event occurs divided by the total number of possible outcomes." (You can have a different definition of probability but this is the math definition.) If you conduct this experiment 100 times with 100 different sleeping beauties, you will get 50 heads flips and 50 tails flips. The 50 heads beauties will be asked 50 questions. The 50 tails beauties will be asked 100 questions. There will be A total of 150 questions asked. Since there are 100 flips of a fair coin, there are no other numbers possible then the ones listed above. Each of the sleeping beauties knows the above. When each of the beauties are woken up they are asked "What do you believe is the probability the coin landed heads?" Each of the beauties knows the coin landed heads 50 times. Each of the beauties knows there will be 150 total questions asked. So from the perspective of each of the beauties the probability that the coin landed heads is 50 ÷ 150 = 1/3. You can look at this any way you want but the above is the one and only way math looks at it.
That is the probability of a fair coin toss, of course. And if sleeping beauty was woken up and asked "what is the probability of a fair coin toss landing heads" she of course will answer 1/2, every time she is woken up, no matter what day it is, and no matter what the coin toss was that decided how many times she was woken her up. But the question she is asked is "What do you believe is the probability the coin flip made after you were first put to sleep landed heads." And she knows if it landed heads she will be asked once, and if it landed tails she will be asked twice. So she know the probability of heads is half the probability of tails. Even though the question(s) she is asked nor her answer(s) does not change the probability of a fair coin toss.
@@jonathanlavoie3115 Well you can reply where/how ever you want, of course. But Marcos' video "Sleeping Beauty Problem Followup" correctly and completely describes the problem, the answers, and the source of the controversy.
Seems obvious to me that the Veritasium fellow is wrongly assuming that asking sleeping beauty "what is the probability of the coin flip being heads?" is the same as if she was to be asked, instead "what is the probability that you have either, just been awakened on a Monday to be fo!lowed by another awakenig on Tuesday, or you have just been awakened on a Monday, not to be fol!owed by another awakening on Tuesday, or you have just been awakened on Tuesday?" But these questions are not the same. Am I missing something?
Thank you for this! This is exactly my thoughts as well. The multiple wake-ups are not independent events. From her perspective she only wakes up once, since her memory is wiped. The *researchers* may be able to use the information of what day she is being awakened to deduce what the coin was, but even so that doesn’t change the probability of the coin flip.
If you know the day, then the conditional probability (given that you know it is Monday, or given that you know it is Tuesday) is indeed no longer 1/2.
P(heads | Tuesday) = 0
P(heads | Monday) = 2/3
But SB is not given any information when she wakes up. She already knew that she was going to wake up whether heads or tails.
It seems thirders tend to accuse halfers of ignoring the conditional part, but they themselves do not recognise that SB does not get any new information. She learns that she woke up, but the probability of waking up is 1 and she knows that.
@@ronald3836Exactly!
Thirders believe that since the SB is awakened, there are more chances she is being awakened one of the two times because of a tails.
But she doesn’t have to know what day it is to have a 1/3 chance. We have Monday Heads, with a 1/3 chance (the 2/3 chance of it being Monday times the 50% chance of it being heads) Monday tails with a 1/3 chance (the 2/3 chance of it being Monday times the 50% chance of it being tails (if heads has a 50% chance and tails is the only other outcome, it also has a 50% chance)) Tuesday Tails with a 1/3 chance (The 1/3 chance of it being Tuesday times the 100% chance of it being tails (if heads has a 0% chance on Tuesday and tails is the only other outcome, it has a 100% chance))
All she knows is that she will wake up and the coin has a 50/50 chance per scenario, so if 2/3 scenarios favor the 50% that is tails, that would give heads a 1/3 chance.
While you are right that she is not given any new information, she still knows that she is either waking up on Monday in 2/3 possible times to be woken up and she is being woken up on Tuesday 1/3 possible times to be woken up.
this can be easily simplified to the following model: imagine there are 100 people sleeping in one room, if the coin is heads then guy #1 is woken up, if it is tails, guys #2-100 are woken up. You are a random person being tested and have no knowledge of what number you are, then are asked the 2 following questions:
1. what WAS the probability of the coin being heads or tails?: it was 50%
2. now that you are awake, what do you think the result of the coinflip was?: you would be right answering tails 99 out of 100 times...
Different problem, this problem takes into account the odds of being person 1 which are low odds
@@samuelhernandez4135 Same low ods of sleeping beauty waking up one time instead of 2. In this example "every wake up" event is a different person and all happens the same day, but ofc you could do heads and tails where SB wakes up 99 times with heads and 1 time with tails... it is just exagerating the probablities for better visualization, as in the monty hall problem
@@pscl_jsc This is my take on it too. Waking up several people gives you the counterargument for when halfers say "it is a fair coin, it is 50/50" as it is more than a fair coin: it is a fair coin AND sleeping beauty was woken up (which she would do twice on tails and only once on heads). However, I would love to see your response to this one: It is the same experiment but the coins is flipped in front of sleeping beauty on Sunday before she is put to sleep. They do not show her what it landed on but they ask her, what is the probability it was heads? She will obviously answer 1/2. Now, the experiment is carried out. When she is woken up, how is it logical for her to now say the coin has a 1/3rd chance of it being heads even though it is the same coin flip she previously called 1/2 chance of heads and she has been given no new information.
@@pscl_jscI can see your point, but I disagree. This model only works in a timeline that everyone exists together, and only one person is awakened. In the original question, a coin is flipped to determine if she would wake up in the heads timeline or the tails timeline. When she wakes up, there's still equal chance that she's living in the heads timeline as the tails timeline
@@pscl_jsc No, the odds are totally different. The person knew the rules going into it, and knew that there was a huge possibility they would not be woken up unless Tails. Being woken up was not guaranteed, and it added information.
I fully agree that there is no paradox here for mathematicians. (Apparently philosophers do find it difficult to understand probabilities.)
But I have a question:
Suppose you wake up and you are told it is Monday. What are now the probabilities?
Then it should also be 50/50, P(Head|Monday) = P(Tail|Monday) = 0.5
@@christoffer1769 Incorrect 😀
P(tails) = 1/2, therefore P(tails and Monday) = P(tails and Tuesday) = 1/4.
Since P(heads and Monday) = P(heads) = 1/2, this gives P(Monday) = 3/4, P(heads and Monday) = 2/3 and P(tails and Monday) = 1/3. (And this is at first sight surprising and perhaps "paradoxical".)
So even though P(heads) always remains 1/2, the conditional probability of heads given what you know might be different.
However, since SB is not given any new information when she wakes up (she already knew she was going to wake up whether heads or tails), P(heads | she woke up) remains 1/2.
@@ronald3836 I agree with your calculations, by saying that: P(Monday and Head) = 1/2, and P(Monday and Tail) = 1/4, Bayes law imply that P(Head | Monday) = 2/3.
I would however say that we have found an expetion where Bayes law does not apply. In this case the right way to assign probabilities is to say P(Head | Monday) = 1/2, and P(Tail | Monday) = 1/2.
Given that it is monday does not give you any information about the outcome of the coin, a "monday-awakening" will occur for any outcome, hence the coin-outcomes should still be given 50% probability each.
@@christoffer1769 by definition of conditional probabilities, P(head | Monday) = P(head AND Monday) / P(Monday) = (1/2) / (1/2 + 1/4) = 2/3. There is no exception to this.
@@ronald3836 I think this example shows that Bayes rule has exceptions.
In this example two different types of uncertainties are mixed together in a strange way; the toin coss which is a "classical" random event, and then the SB's lack of knowledge (a completely different type of uncertainty) which forces her to divide her believe (i.e. divide P(Tail)=1/2 into P(Tail and Monday) =1/4 and P(Tail and Tuesday)=1/4)). It is no reason to believe Bayes theorem should handle this strange situation.
And further, if one is a halfer and accept the "no new information"-argument, it would be contradictory to suddenly take a "thirder view-point" and assume monday gives you new information. Monday gives no new information about the coin toss, just as waking up gives no new information.
Your problem with an answer being reached by means of consensus is justified. Obviously, an incorrect answer is not made correct by receiving a majority vote. (With one exception… Democracy. You’d be surprised how many people will accept something as fact, just because most of the people around them voted as such.)
However, taking issue with an answer being reached by consensus in this situation is irrelevant.
He did not suggest that a correct answer would be reached by voting. He didn’t imply that there was even a decision to be made. He only asked for people to vote for which answer they liked or agreed with the most… that’s all.
You assumed that he has an intended use for the votes, since he made no such claim.
That being the case, the correct statement would be to say that you have a problem with your own assumption that he intends to base a claim of a conclusive answer on the votes he receives…
Which is not a very good way to begin a rebuttal video… and we’re only 39 seconds into your response.
I kind of feel like most of the halfers' reasoning ultimately relies on "bro lmao it's a fair coin you *know* the chances are 1/2H and 1/2T. Stop confusing yourself with random bullshit", but man, *every* person who tackled the SB properly is smart enough to understand the meaning of "fair coin". We all get it. The unsettling thing about this problem is that bayesian reasoning *still* seems to suggest that, from the SB perspective, every time she's asked the question she is more likely to be in a "the coin landed tails" scenario.
I'll basically copypaste the wikipedia page for this, since it's so well done. Let's assume that the sleeping beauty, when woken up, was *told* that the coin landed Tails (so it could be either Monday or Tuesday - she doesn't know, of course). Now a crucial part of the reasoning: would you agree with me that, in this scenario, to the eyes of the SB, it would be equally likely for the day to be either Monday or Tuesday? If the answer is yes, we can write down the following equivalence between conditioned probabilities:
P(Monday|Tails) = P(Tuesday|Tails)
That is, the probability that today is Monday (assuming we know that the coin flipped Tails) is equal to the probability that today is Tuesday (assuming we know that the coin flipped Tails). The conditioned probability of an event P(A|B) is basically just the probability of both A and B happening, but "renormalized" on the probability of B happening (because we know that B happened) : P(A|B) = P(A and B)/P(B).
We deduce that P(Monday|Tails) = P(Monday and Tails)/P(Tails), so that in the equivalence found before we can simplify a P(Tails) to get:
P(Monday and Tails) = P(Tuesday and Tails)
We now reason similarly to before: let's assume that it was *told* to the SB that today is Monday (but she still doesn't know the outcome of the coin toss). Crucial assumption #2: would you agree that, in this scenario, to the eyes of the SB, it would be equally likely that the coin either flipped Heads or flipped Tails, since the Monday wakeup is scheduled in either case? If the answer is yes, we deduce:
P(Heads|Monday) = P(Tails|Monday)
Again, we simplify a P(Monday) to the denominator, and get:
P(Heads and Monday) = P(Tails and Monday)
But we also already found that P(Monday and Tails) = P(Tuesday and Tails) (of course, P(A and B)=P(B and A), since they both represent the probability of both events happening together), so that we deduce:
P(Mon and H) = P(Mon and T) = P(Tue and T)
In other words, _to the eyes of the SB, every time she wakes up these three events are all equally likely_ . Since in two of these equally likely scenarios the coin flipped Tails while only in one it flipped Heads, we deduce that the best answer of the SB based on its priors and assumptions is the thirder position.
*Yes*, it is true that all the waking up events are not independent from each other - in fact, if you are woken up any day outside Monday the probability of Tails is 100%. But the whole point of the SB setup is that _she has no idea which day is today_ , nor how many times she's already been asked the question. This corresponds to a lot of information being lost, and if you run the maths and assume that Bayesian probability works, you end up in the thirder position. It's true that if it comes up Tails you already know that the SB will be woken up two times, but to her every awakening is a clear new experience: she has no way to distinguish neither of them from the single wake up caused by the Heads outcome, and she has to weight this in when making its guess on the outcome. This is reflected in the first crucial assumption I made: even if the SB was told that the outcome is Tails, thinking "Oh, it's easy! I know they'll wake me up two times in total!" doesn't give her full intel of her situation; she's still stuck with "But wait... Did they already wake me up, or is this the first time?".
Let's put it this way. The thirder position is the correct answer to the question: "what is the probability that today, right now, the situation you're experiencing was caused by the fact that the outcome was Tails?", Which to me is very similar to the question: "What is the probability that the coin came up Tails?", maybe with a subtle "[given that you're awake right now?]" implied (which, together with the lack of information regarding the current day and past days, is the bit that causes all the trouble).
If you accept the two assumptions that I labelled as "crucial" to be true, there's little escape. In a bayesian analysis of the problem, the Tails outcome simply weights more in the mind of the sleeping beauty - because the experiment conditions don't allow her to catch the blatant correlation between being woken up 2 times and the coin flipping Tails. Every day weights the same, since she doesn't know if it's a Monday or not.
She is asked what the probability is. From her perspective. The information that she does know is 1. Coin is flipped. 2. She is woken up on each day. That's all she knows. So the question from her perspective (the actual question). There are implications (as stated) like "she should". Why should anything change if she has no new info. From her perspective Tuesday is no different than Monday to her. And the question is "from her perspective".
You misunderstood the question. SB is asked about her credence that the coin came up heads (at a known past time). She is not asked about the expected behavior of a fair coin.
So it's a problem in forensics: She must consider the evidence, in addition to the physical behavior of a flipped coin. The physical behavior gives her prior credence -- her expectation of the outcome of an arbitrary flip of the coin. Put in odds form, this is 1:1 for a fair coin.
But she isn't asked about an arbitrary flip. She is asked about one particular flip, the one that happened Sunday night. And that flip has consequences: If it is tails, she'll be awake on Tuesday; if it is heads, she will not be awake on Tuesday. Therefore, if she finds herself awake on Tuesday, she knows it was tails. If she finds herself asleep all day Tuesday, it was heads, although she isn't cognizant in this case.
Unfortunately, it's not that easy: She doesn't know, when she is awake, if it is Monday or Tuesday. So she must apply Bayesian reasoning. "What would be the probability of my being awake on an unidentified day during the course of this experiment, if the toss was heads? What would it be if the toss was tails?" The number that matters is the ratio of these two values, which is 1/2. This is called the likelihood ratio.
In Bayesian reasoning, you multiply the prior odds (1 to 1) by the likelihood ratio (1/2) to get the posterior odds (1 to 2). Odds of 1 to 2 converts to a posterior probability ( SB's *credence*) of 1/3.
She wakes up. She knows she would wake up with heads, and she knows she would wake up with tails, so both are equally possible: prob 1/2.
She also knows that if it was heads, then it is Monday.
And if it was tails, then it is either Monday or Tuesday, both with prob 1/2.
So P(Heads and Monday) = 1/2, P(Tails and Monday) = 1/4, P(Tails and Tuesday) = 1/4.
This means P(Heads | Monday ) = 2/3, P(Tails | Monday) = 1/3. At first sight this may be counterintuitive. But still it is right.
@@ronald3836 Why?
There is exact math (Bayes's Theorem) that shows she should have a 1/3 credence in heads. She is asked about her credence in a past event, not about the expected behavior of a fair coin. It's not a forecast, it's a forensics problem. So she must take into account all of the clues at her disposal, including the fact that she's awake. She's twice as likely to be awake on a given day of the experiment if the toss was heads.
You misunderstand the Bayesian reasoning. In order for the apriory hypothesis to change, additional information must be provided. In this quasi-problem, no information is provided. She might as well be answering before waking up, as she has the same amount of information then as she does afterwards, that being that she is going to wake up. period. This is a pointless trick question that uses the arbitrarily assigned weight to one of the events making you think it will become more likely. But this is both a mathematical and a logical phalacy. While it is true that there are twice as many opportunities for her to be right by going with tails, each of those only has a 25% chance of occuring for her. However if you view it from another angle, if tails happnes she is guaranteed to get both of those scenarios, wereas heads only gives her the one. So if she wants to be right as many times as possible, she should pick tails, but that is just like saying that because rolling a 6 gives you 1000$ while other numbers get you only one, you are more likely to get a six.
But this question is just vague enough to make both answers seem plausible. This is why you dont mix math with philosophy, the correct answer according to probability theory is 1/2.
@@RealAXork *a priori*
By definition, an a priori assumption does not change, but let's gloss over that. I don't see how your argument can resolve the Technicolor Beauty problem described by Michael Titelbaum:
"Everything is exactly the same as in the original Sleeping Beauty Problem, with one addition. Beauty has a friend on the experimental team, and before she falls asleep Sunday night he agrees to do her a favor. While the other experimenters flip their fateful coin, Beauty's friend will go into another room and roll a fair die....If the die roll comes out odd, Beauty's friend will place a piece of red paper where Beauty is sure to see it when she awakens Monday morning, then replace it Tuesday morning with a blue paper she is sure to see if she awakens on Tuesday. If the die roll comes out even, the process will be the same, but Beauty will see the blue paper on Monday and the red paper if she awakens on Tuesday.
"Certain that her friend will carry out these instructions, Beauty falls asleep Sunday night. Some time later she finds herself awake, uncertain whether it is Monday or Tuesday, but staring at a colored piece of paper. What does ideal rationality require at that moment of Beauty's degree of belief that the coin came up heads?"
-- Titelbaum, "Quitting Certainties", 2013
She gains no information from the color of the paper as to what day it is, or as to the result of the coin toss, since the results of coin and die are independent. Yet a Bayesian analysis shows that, given she sees a red paper, she must assign a 1/3 credence in heads; likewise, she must assign a 1/3 credence in heads if she sees a blue paper.
If she is not allowed to update based on the color of the paper, then she will have the same credence between the moment of awakening and the moment she sees the paper. So she must already have a 1/3 credence in heads as soon as she wakes up, which is analogous to her waking up in the original experiment.
@@rsm3t
I am bad with terminology and spelling as you've pointed out, but ignoring that, I don't see how the die introduces anything new to the problem. It is once again a matter of what the question is really asking. If she really wants to know what the outcome of the toss was she should look ak it probabilistically. She knows that the odds of the coin coming up heads are 50% just like the die rolling an odd number. She also knows they are independant events so she is just as likely to find a blue paper as she is to find a red one and this tells her nothing about the outcome of either event. So once again, she gains no information and can't update her hypothesis. But if she wants her answer to be correct more times than not, than she should always bet on tails, since that gives her more oportunities to answer. That is a different question though. But from a philosophical standpoint it's valid to ask which of these two questions she should be asking. If you ask me though, it's kind of a pointless debate.
You made very logical arguments. Smart analysis on your part!!!
7:30 he agrees with you that the probability doesn't split, he's just saying that as an example of what someone might initially think
The question asked to Sleeping Beauty is "What do you believe is the probability the coin landed heads?"
The question is not asked to you, an outside observer.
The question is not "What is the probability of a fair coin landing heads?"
The question is not "Did the coin land heads?"
The question is not "If today is Tuesday what is the probability you were asked this question before."
All of those question have the exact same answer. It is 50% chance.
Marcos' video "Sleeping Beauty Problem Followup" has the correct, exact and complete explanation to this problem.
@@tedr.5978 he's hopeless; he doesn't understand what is the problem even after he saw the followup... and also continues to like his own comments 😂.
@@tedr.5978 Yeah you mean an explanation where he somehow ends up with more balls than there existed at the start? Among other things, Leftism is an ideology of conjuration magic. The Leftists believe they can conjure stuff out of nothing. Exactly like witches do. And lo and behold, some of the highest ranking leaders of the Leftist cause call themselves, you guessed it, witches!
I rest my case.
Exactly, thank you! This is the correct solution.👍
It would be like choosing between two different bags, one with one white marble and one with a million black marbles
Sorry for the imperfect English it is my second language but if you have any questions I would love to clarify and if you want to provide counterarguments to any of my claims, I would love that even more. Stay respectful
Good job Marco. There are two outcomes to the coin toss. If the coin comes up Heads then Sleeping Beauty is woken once. If the coin comes up Tails then Sleeping Beauty is woken more than once. The probability of each of these is 1/2.
If the outcome is Heads or Tails then SB is woken Monday. The probability of this awakening is 1/2 in each case. However in the case of Tails she is also woken on Tuesday. The probability of the Tuesday awakening by itself is 1. It is certain. It is dependent on the Tails. If the toss comes up Tails then she will be woken Tuesday. Therefore the probability of SB being woken Monday and Tuesday is 1/2 x 1 = 1/2, the same as the Heads outcome.
The mistake people make is to treat the Tuesday awakening as having the same probability as the Monday Head or Tail awakenings. It does not. It is a certain result of a Tails outcome with no probability that it not occur.
On awakening, SB would realise she does not know when she is being woken or how many times she has been woken previously or will be woken in the future. It may seem that her awakening is one of three possible outcomes, Heads Monday, Tails Monday or Tails Tuesday. Casual observers we might be tempted to split the probabilities equally. However SB might realise that a Tails Monday and Tuesday awakenings are a single result of a Tails outcome in the coin toss with the same probability as a Heads awakening outcome.
No matter how many times she is awakened after a Tails outcome the probabilities are the same. That is, more awakenings do not make it more probable that SB is waking up after a Tails. In other words every awakening is the outcome of a single toss of the coin.
Hope this helps.
Each wakeup has a 1/2 chance to happen, but as they are equally likel to happen, a 1/3 chance to be the one you are experiencing. So given you wakeup, the chance is 1/3 that you happen to be wokenup while the coin toss was heads.
@@OzoneTheLynx Let's say I flip the coin and let it land on the floor inside a vault where I lock it for 10 years. And then I put her to sleep. The physical event of the coin flip happened 10 years ago. It does NOT matter how many days I arbitrarily choose to wake her up if the coin lands on the tails.
@@OzoneTheLynx Chance is not 1/3 each. You have a 50% chance of experiencing the heads wakeup and a 25% chance of experiencing each of the tails wakeups (as your current wakeup).
@@hypersonicpiano6120 No.
Me: 1+1=2
@@kaizokujimbei143: no
*Consider another experiment:*
A coin is flipped and if it is Heads you are directed draw a marble from bag A that contains 5 White marbles and 5 Black marbles. If it is Tails you are to draw a marble from bag B that contains 9 Black marbles and 1 White marble.
The experiment is run: The coin is flipped and the result of the coin flip is concealed from you. You are presented with a bag and directed to draw a marble from it. You know that the bag presented to you is either Bag A or Bag B but since the result of the coin flip was concealed from you, you are unsure which bag you are drawing from.
After running the experiment you ended up with a Black marble. What is the probability that Heads was the result of the initial coin flip?
*Discuss.*
5/14?
@@submanstan7488 I guess this is how a thirder would reason:
It is less likely that the coin came up heasds because the black marble is more likely to have come from bag B (which contains more black marbles).
Expressed mathematically:
P(heads) = 0.5
P(black | heads) = 0.5
P(black | tails) = 0.9
P(black) = 0.5*0.5 + 0.5*0.9 = 0.7
P(heads | black) = P(black | heads)*P(heads)/P(black) = 0.5*0.5/0.7 = 5/14
A halfer would reason as follows:
The coin is fair and nothing else matters, thus the probability is 0.5.
Expressed mathematically:
P(heads | black) = P(heads) = 0.5
Reading it again@@Minkouski I would change my answer.
Since the coin is only flipped once and a coin only has two options (excluding landing on its edge) I would now go with 0.5.
The whole 5/14 thing is a red herring. We all know that doesn't make any logical sense.
Multiple coin flips, however, would change things.
Your explanation here is correct, but for a slightly different reason. The reason why original video is wrong, is because they are answering a different question than they are asking. The question of: "What is the probability that the coin came up heads" has the correct answer 50%. But the question that the original video is actually answering is: "what is the probability that the sleeping beauty is correct when she is woken up and asked "why did we wake you up?" and she states that the reason is that coin came up tails?". The answer to this question is 66.67%, because she knows she is 2x as likely to be woken up from tails then heads. So if she says tails all the time she will be correct 66.67% of the time.
This is the canonical form of the question: "What is your credence now for the proposition that the coin landed heads?"
*your credence now* means "your credence, knowing that the experiment is in progress and you have just woken up".
the probability that she is correct on any given ask is 50%
you’re making the mistake of assuming that the wake-ups are equally probable
He didn’t use the like button to say what the right answer IS, its just a creative experiment
Appeal to Authority - sheep.
more than half do.
Initially I answered 1/2 however one compelling argument I can see for the thirder position is to consider a slightly different scenario, in which if it's Tuesday and the coin landed on heads then she is still woken up, but within 10 seconds of being woken up she is told that it's Tuesday and the coin landed on heads, while otherwise she is told nothing. In this new scenario when first woken up there's a 1/4 chance that she will be told that it's Tuesday and the coin landed on heads, and once 10 seconds have passed if she has been told nothing then she can eliminate the possibility that it's Tuesday heads and the probability of the remaining possibilities remain equal to each other, and so go to 1/3.
Thinking about the original scenario, I would argue that during the times that she is woken up, it's always at 12:00 noon that she's woken up then at 11:59 there's a 1/4 chance of it being Tuesday and that the coin landed on heads, in which case she will not be woken up at 12:00 noon. If it's 12:00 noon and she hasn't been woken up then the probability that it's Tuesday and the coin landed on heads is 100%, however she isn't conscious to consider that it's 12:00 noon and she hasn't been woken up. Still I would argue that she can eliminate the 1/4 chance that it's Tuesday and the coin landed on heads from the possibility space as soon as she wakes up, and so being awake, even without being told information does convey information in the sense that it's not Tuesday Heads and so change the probabilities. When she's asleep there's a slightly higher probability that it's Tuesday and that the coin landed on heads as she could be sleeping when she would otherwise have been awake, although she isn't conscious to consider that.
This clinches the disparity of probability assessment: 16:21 "They are two independent probabilites" The second probability calculation has an added variable. Sleeping Beauty must factor in, not just the coin toss, but also the days such that Tails is weighted by 2 days versus Heads with just 1 day.
It's a question involving the sleeping beauty's first-personal beliefs and the amount of credence she has in assingning probabilitites to the coin flip. The question is not about the probability of the coin turning up heads from a third-person point of view, as you are assuming.
Halvers and thirders answer to 2 different questions. One is "what is the objective probability of getting H or T?", the other is "What is the probability that the sleeping beauty will correctly guess whether it was H or T every time she is put out of sleep?". The sleeping beauty can either give the same answer based on a logical conclusion (skewed probability of guessing H or T) or answer randomly as she can't remember anything. So if we do this experiment for just one week and she answers the question randomly, the chances are on average 50:50. If she sticks to one answer she will either be right 2 times / wrong 1 time or right 1 time / wrong 2 times. If we repeat the experiments for many weeks, from the guessing statistics perspective it's better for her to always bet on T.
Say I draw a card from a standard deck of playing cards. I don't show it to you, but I tell you it is a black card. If I then ask you for the probability that I drew the Ace of Spades, am I asking you for the "objective probability of drawing the Ace of Spades (1/52)" or "The probability of guessing correctly that this card is the Ace of Spades (1/26)?"
Neither is correct, although the second is closer. The first is laughably absurd. The question is clearly posed within the state of knowledge that the card is one of 26 black playing cards, not one of 52 playing cards of either color. To make a similar claim about the coin is equivocation, trying to support an argument that is not actually the one used by experts who are Halfers.
The second is projecting a philosophy of assigning values to probabilities, called frequentism. I must stress that it is a philosophy, not a mathematical definition. But it does not apply here since the number of trials depends on the result.
The actual question is unambiguous: within the state of knowledge that you have at this time, what is the probability of Heads? Halfers argue that this state is no different than the state, on Sunday, that includes another possible awakening. Thirders argue that each potential waking is a different state when viewed from within it.
@@jeffjo8732 , your example is not analogous to the Sleeping Beauty problem. She does not acquire any new information during the experiment and therefore her state of knowledge remains the same during the experiment. However, when you tell me that you drew a black card you give me a lot of new information.
Maybe I can clarify this with another example. Suppose that I have two deck of cards, one of which contains only red cards and second contains only black cards. I pick one of the decks, but you don’t know which. I draw a card. What is the probability that I drew the Ace of Spades?
@@user-yx1oj1fs9b The example was to show how to interpret the question, not how to answer it. The point was that "Halvers and thirders answer to 2 different questions," as stated in the comment I replied to, is completely wrong. They are answering the same question, in the same context, but disagree about what that context means. Thirders understand what "new information" is, and know that SB has it. Halfers do not; they never even define it, they simply assert that it doesn't exist.
Your "clarification" is completely irrelevant. It is a straight-forward probability problem, not a conditional one. But assuming that the black deck has N cards and M Aces of Spades (you do realize that you didn't specify this, don't you?), the answer is M/2N.
Anyway, the prior probability space for any probability experiment includes a sample space; that is, a set of all disjoint outcomes of that experiment. The mistake halfers start with is thinking that the sample space is {H,T}. Since SB's state of knowledge can only ever span one day, the correct sample space is (Mon,H), (Mon,T), (Tue,H), (Tue,T)}. What halfers further don't understand, is that (Tue,H) is an outcome, that SB knows it is an outcome on Sunday Night, whether or not SB understands that she can't actually observe it when it does. But when she is awake, she knows that (Tue,H) is not the outcome she is observing. It can be any of the other three. THIS IS WHAT "NEW INFORMATION" MEANS. She has it, and can use it, and the answer is 1/3.
This might be clearer if the testers tell her this: "You will wake up in one, or two, different situations; a fair coin toss will determine which (Heads=1, Tails=2). But we will wipe your memory and put you back to sleep after each, so you will not be able to tell if you have been awakened before. Essentially, each waking will be an independent outcome of this experiment within your state of knowledge. While you are awake....").
There are other ways to prove that 1/3 is correct, but halfers look at them with the same blind eye that they use to claim that (TUE,H) cannot exist.
@@jeffjo8732, thank you for your response. Of course, you are right about my counterexample being underspecified, but it is just fairly simple one and I (rightfully) assumed that you will figure it out easily, which you did. I’ll try to explain what I meant. First of all, I was responding to your cards example which I think was drastically different from what is going on in the SB problem.
Suppose I split the regular deck of cards into two packs, one of which contains 26 black cards and another contains 26 red cards. Then I pick one pack without you knowing which one I picked. I decide which pack to pick by flipping a fair coin. Then I drew a card out of this pack and ask you what is the probability that the card is the Ace of Spades. Your answer would obviously be 1/52 (M/2N in your notation for a general case). And this answer is essentially a statement about how the world really works.
Funny thing about it is that this answer is sort of absurd since at that point you know for sure that the probability is either 1/26 (if I picked the pack of black cards) or strictly zero (otherwise). Of course you may argue that since you don’t know which pack I have the probability of getting the Ace of Spades is 1/26*1/2+0*1/2=1/52. But if you want to maximize your chances of being right answering the question you have to answer that the probability that I drew the Ace of Spades is zero precisely because you know that I either picked red cards or black cards with the probability of 0.5. Going with zero you have 50 % chance of being right. How so? Well, because you are not trying to make a claim about how the world really works but rather trying to predict as accurate as you can the outcome of this particular attempt given the information you have. And this information is this: you know, that I picked a particular pack of cards. Therefore, there is a 50 % probability that I’ve picked the red pack, and the probability of drawing the Ace of Spades from the pack of red cards is zero. That’s the crucial point: zero is not the wrong answer in and of itself. In fact, it is exactly the right answer for those cases in which I pick the red pack. It is just not right generally. In other words it is the right answer to the different question.
The difference if that in one case we are talking about actual probabilities (about how the world really works) and in the other case we are talking about our credence for some particular propositions about how the world really works. If you just draw a card from a regular deck the probability of getting the Ace of Spades is 1/52. What is my credence for a proposition that instead this probability is 0? Well, it is zero, I am 100 % confident that it is not 0. If you follow my procedure and split the deck into two halves and then draw a card from one half of the deck what is the probability of getting the Ace of Spades? Well, it’s still 1/52. But what is my credence for a proposition that instead this probability is 0? Well, 50 %. I am 50 % confident that after you split the deck the actual probability is 0. It is the same with the SB.
You yourself stated in the original comment that “The actual question is unambiguous: within the state of knowledge that the SB has at this time, what is the probability of Heads?“. Well, does the state of knowledge that the SB has at any given time include the knowledge that the probability of fair coin landed heads is 1/2? If so (which is the case in the SB problem) why would any additional information alter this? If the SB already knows how the world works (in this case she knows that fair coin lands heads or tails with probabilities of 1/2) she already has exhaustive state of knowledge. However, if we are talking about her credence in some particular outcome of experiment, it is different. She doesn’t know whether the coin landed heads or tails but she knows that she is twice as likely to be asked about it if the coin came up tails. So her credence for that scenario is naturally 2/3, and her credence for the coin came up heads is 1/3. But these are not her estimates of the probabilities, rather her credence for different scenarios of the experiment. In exactly that sense halfers and thirders are answering the different questions. Halfers answer the question “What is the probability of the coin came up heads?” Right answer to this one is ½. Thirders answer the question “What credence the SB should have for a proposition that the coin came up heads?”. Right answer to this one is 1/3.
@@user-yx1oj1fs9b "I was responding to your cards example which I think was drastically different from what is going on in the SB problem." Again, the "cards example" was not meant to say anything about "what is going on in the SB problem." It is meant to show that asking for a probability, after establishing a state of knowledge, is asking for the conditional probability within that state of knowledge. That's all.
"Funny thing about it is that this answer is sort of absurd since at that point you know for sure that the probability is either 1/26 (if I picked the pack of black cards) or strictly zero (otherwise)." This is a fallacy. You are saying Pr(Ace of Spades|Black)=1/26 and Pr(Ace of Spades|Red)=0, but ignoring that you are valuating conditional probabilities.
So if you consider it seriously, you don't understand probability at all. The exact same logic says that the probability is either 1, or 0. The card either is, or is not, the Ace of Spades. The point of probability is to measure the relative possibilities of a set of outcomes that are all possible. You can narrow the set based on knowledge that the set represents the set. And it works the same way whether that result has not yet been determined, if it has but you don't know anything about it (as you claim describes the SB problem), or you have partial knowledge (as is the actual case).
"Does the state of knowledge that the SB has at any given time include the knowledge that the probability of fair coin landed heads is 1/2? " More accurately, it includes the assumption that a fair coin, *_when_* *_it_* *_was_* *_flipped,_* had a 50% chance to land on Heads. But it also includes the knowledge that there were four possible situations that follow from that flip, depending on Heads or Tails, and some are different than what is happening.
She has no information, other than the schedule, to determine which of the four represents her current situation. The halfer argument starts with the assertion that is essentially that one Heads situation can never exist. Being unobservable is not the same as having no possibility to exist.
It goes on to argue, incorrectly, that the two Tails situation are the same situation, since they are part of the same future on Sunday Night.This is absurd - they are distinct situations, and SB is in only one of them. What if we wake SB on Tuesday, after Heads, but take her shopping instead of interviewing her? The argument that (Mon,T,Interview) is the same outcome as (Tue,T,Interview) also says that (Mon,H,Interview) is the same outcome as (Tue,T,Shopping). Like I said, absurd.
Finally, she is not asked for the probability that a fair coin, *_when_* *_flipped,_* lands on Heads. She is asked for the probability, given her current situation that depends, in part, on a (past tense) coin flip, *_landed_* on Heads. These are not the same thing.
That's why "additional information alters this." Because the random experiment is not just the coin flip, it is everything that leads up to SB being awake and interviewed. That includes dividing the observation opportunities between Monday and Tuesday.
Veritasium (5:03) "She learns that she's gone from existing in a reality where there are two possible states, the coin came up either heads or tails, to existing in a reality where there are three possible states."
Marco (5:24) "This is exactly what I was saying. They are not three states, they are only two states but tails is split up into two sub-states. But these two sub-states are not equivalent to the one heads state or the overall tail state."
Both are wrong. Marco's position makes the common error that connects all halfers. The Veristasium position is a misrepresentation of Adam Elga's original thirder argument, which is closer to Marco's but avoids the significance of that error.
There are four states (or two sets of two sub-states, which is the exact same thing). The people who want to argue about epistemology (the theory of the scope of knowledge) try to convince us that Tuesday, after Heads, somehow ceases to exist because SB sleeps through it. It can exist, SB knows it can exist unobserved, and SB knows that what she is observing is not that state (or sub-state). So of the four equally-likely states (or two equally-likely states, each with two equally-likely sub-states), SB knows that she is in the only one of three observable states where the coin landed on Heads.
And there is an easy way to demonstrate how the "missing" state (or sub-state) must be present in her knowledge. There are actually four different, but equivalent, ways that the experiment can be performed. The day when SB is always wakened can be Monday (as in the video) or Tuesday (by exchanging the procedures on the two days). Then, the option to let her sleep can come after Heads (as in the video) or Tails (and she will be asked about Tails). Each way that it can be performed must have the same answer.
So, use four volunteers. Assign a different one of the four possible ways to each, but don't tell them which. Use the same coin flip for all four. On both Monday, and Tuesday, three volunteers will be wakened. Bring them together, and ask them, as a group, to assign a probability to each individual, to the proposition that this is her only waking.
Here, each of the volunteers can be considered to represent one combination of (Coin, Day). Three "exist" in the sense that they are in the awake group, and the fourth "exists" in the sense that they know about her, and know that she is asleep. They also know that exactly one, of the three who are awake, will only be wakened once. And that there is nothing that allows them to say that the probability of being that one is different for any of them.
That probability is 1/3.
I explained to a dozen 1/3 ers why the answer is blatently obvious to be 50/50 snd no amount of logic or reasoning changed their mind.
It is scary to me that these people might be on a Jury trial, or otherwise affect someones life.
No, what is really scary is that you think the calender skips from Monday to Wednesday if we don't wake Sleeping Beauty. There are four possible states where we have to decide whether to wake SB. One is _eliminated_ as a _possibility,_ not _removed_ from _existence,_ based on the evidence that SB is awake. When outcomes are eliminated based on evidence, but still exist in the experiment, Mathmatics tells us to update the probability to:
P(H|Awake) = P(H&Mon)/[P(H&Mon)+P(T&Mon)+P(T&Tue)] = 1/3.
Umm, you're joking right? I think you need to re-watch the video lol.
@@andyh8239 You're joking, right? When SB imagines, on Sunday, what Tuesday will be like, is it:
1) There are four distinguishable (not to me, to the experiment), equiprobable states of the experiment that I could exist in over the next two days. It can be Monday after Heads, Tuesday after Heads, Monday after Tails, or Tuesday after Tails. I won't be aware of Tuesday after Heads, BUT THE EXPERIMENT STATE EXISTS REGARDLESS OF MY WAKEFULNESS.
2) The states Monday after Tails and Tuesday after Tails will exist at the same time, since my conscious memory will experience both but not be able to distinguish them. The state Tuesday after Heads cannot exist this way, since it is distinguishable from Monday after Heads be if it does exist.
The answer to the question is (# observable states with Heads)/(# observable states). That's 1/3 in interpretation 1, and 1/2 in interpretation 2. In fact, the only way to get 1/2 is interpretation 2. Most Halfers understand the "Monday same as Tuesday" part, yet somehow accept it. They don't realise that the "Tuesday ceases to exist" case is required as well.
If you doubt me, change the problem so you wake her on both days, but don't ask for a probability on Tuesday after Heads.
There are other, more formal ways to prove the answer is 1/3. Halfers ignore them like Trumpers ignore stolen documents.
The sleeping part is irrelevant, asking someone the probability of a coin flip twice in a row does not change the probability of the coin flip.
But it changes the probability of guessing the result of the coin flip correctly. In this case there’s just a probability of guessing twice. But even just a probability of guessing twice will change the probability of guessing correctly.
@@laowei7279 No. She's not being asked that. She was asked what the odds of a 50/50 probability were.
It doesn't matter what day it is, what actually happened, or anything else. The odds were 50/50 period.
Even if it IS Tuesday the probability didn't change.
My take: Given that I sleeping Beauty am awake, and given that if the coin lands tails I will wake up a million times, it is more likely that tails has come up, but if the question is, given that I am awake, which SET (the set of one wake versus the set of plural wakes) am I in, then it’s equally likely in each every instance of waking that the coin was either heads or tails. Subtlety different questions. Am I wrong?
Well that set should have an equal probability to the sum of its compoments since they are disjoint events. So if you think the set has a total probability of 1/2 it makes no sense for those million wake ups to add up to any more or any less.
What I think confuses people is the fact that you are guaranteed to wake up all of those times which makes them feel like waking up is more likely, but what you forget is that due to the memory wipe, each of those wake ups is entirely dependant on the others and only one can occur from beauties perspective (she forgets the others). This means that the probability of her waking up for one of those days is 1/2n where n is the number of days she is going to wake up for tails, and 1/2 for waking up once for heads.
Mathematically, there is no paradox. The two answers are correct depending on how the question is interpreted.
However, I believe that there is something hidden here. Mathematics is an abstract view of reality. Reality is much more complex. I do believe that there is an opportunity to introduce a new concept that clearly defines the difference. A new mathematical symbol or equation that describes the change. This could have a true impact of our understanding of reality.
You are correct in one way. If interpreted "when it was flipped, what was the probability it came up Heads?" then the answer is 1/2. And if interpreted "based on your state of knowledge now, what is the probability it came up Heads?" then the answer is 1/3.
But you are wrong that the first interpretation is a valid one. Say I ask you to draw a card, but only let me look at it. If I tell you that it is an Ace, is the probability that you drew the Ace of Spades 1/52, or 1/4?
Both questions clearly want you to use the information you have. True Halfers think that information does not allow you to say the answer changes between the two interpretations.
100% agree
Not trying to hate but this video (and a lot of the comments) seem a bit nitpicky to me regarding what question Derek was asking.
Your tone and the comments strike me as a little condescending which is a shame. I would have loved to watch a video that makes good/ logical points and maybe even gives a definitive answer to such a controversial problem.
But since your other viewers and commenters don't indicate to feel similarly you are well within your rights to ignore this comment I guess. (:
But if you'd like some constructive criticism in case you want to grow your community and not have first time viewers like me leave when they click on your videos: Maybe try to use more auxiliary verbs and even if you're fairly sure that your conclusion is right, treat it as an optionon. That makes it easier to stomach for people that might disagree with you thus they won't click away immediately.
Also, I thought it came across as a little mean (although you probably intended it to be funny) when you edited 1:56-2:18 and asked sarcastically "so what is the problem here". If the point of your video was to argue that it doesn't matter when you ask an individual in the 'sleeping beauty' role what the probability is then this video could have been a lot shorter. Instead, you picked apart evrything Veritasiums' video argued with the same argument which unfortunately came across as very spiteful.
I hope you don't see this as a personal attack and whether or not you choose to agree with me or not, I hope you have a good time on youtube.
It was just too smarmy for my taste, you can give your response without the "looping the video with slow-mo" bit and using a condescending tone - what an unlikable response
This is utterly frustrating. Stop trying to rephrase the question. We are asking Sleeping Beauty if she thinks the likelihood she's been awoken by a heads or tails flip.
Maybe he should take his own suggestion and rewind, and listen again. It IS NOT asking the likelihood of a coin landing heads or tails.
Now, with fresh perspective, if you're woken up twice each time the flip is tails, can you not see the other argument that from her pov, heads has only woken her up 1/3 the time?
What...😂
Ok then riddle me this: If the coin lands heads, she is neither woken up on Monday or Tuesday, but if it lands tails, she wakes up on Monday and Tuesday. Would you still say that she should say "there is a 50% chance the coin landed on heads" in this case? After all, that is the probability of the coin.
In this case waking up gives you information, namely that you woke up. The conditional probability of tails given that you wake up is 1.
In the original problem, waking up does not give you any new information. But you are right that this is important.
Puzzle: in the original problem you wake up and you are told it is Monday. What are now the probabilities?
@@ronald3836 "Puzzle: in the original problem you wake up and you are told it is Monday. What are now the probabilities?"
also 50% each. David Lewis is a fake halfer.
@@Feds_the_Freds I realised my modification does not result in a well defined question. By telling SB the day, you change the rules of the experiment. If she understands that the rule is now that she is always told the day, then the probability of heads when she is told it is Monday is 1/2. But if she assumes she is not supposed to know the day but by a fluke event gets to know that it is Monday when she wakes up, then she should say 2/3.
@@ronald3836 what do you mean by it being a fluke? Something like: She isn't supposed to know the day, but finds it out anyways? how would you well define that?
Maybe something like: the researchers toss a parallel coin and tell sb the day, if that parallel coin lands heads.
So P(Heads|Heads AND Monday) = 25%
P(Tails|Heads AND Monday) = 25%
hm, still 50/50.
I agree that she should say that it's 2/3 heads given that she finds out the day (Monday) by a before unknown event but how to construct the experiment?
Do we tell sb that it's the normal sb problem but what we actually do is "spoonfeed" her the information on monday anyways, which day it is? Like, a reasearch assistant goes in and says "have a good monday evening" and the interviewers are seemingly angry that he gave sb additional information that she wasn't supposed to know?
Is that modification to the experiment actually what David Lewis meant?
@@Feds_the_Freds yes, that is what I mean. I'm not sure how to define it properly. But you could think of a scenario where unexpectedly a siren goes of on the first Monday of the month (in my country sirens used to be tested on the first Monday of the month), which tells SB it is Monday (or war broke out). SB will understand that this was not an intended part of the experiment. I think now she should estimate the probability of heads to be 2/3. Before she heard the siren, it was 1/2 for H on Mon, 1/4 for T on Mon, 1/4 for T on Tue. After the siren she knows it is not T on Tue, so the other two become 2/3 and 1/3.
But this is really subtle and perhaps I am wrong. If instead of the siren there is an envelop that she can open to learn the day (and she knew this envelop would be there), and she has to make guesses before and after opening the envelop, then I think she should say 1/2 before opening and again 1/2 after seeing it is Monday.
From her perspective, it doesn't matter the actual coin probabilities, but the probability of being awakened with a certain result in the last coin toss. If the experiment is repeated many times, she will be awakened following heads twice as much as following tails. If she receives a dollar every time she guesses the result correctly, she should answer heads every time.
Did you get that the right way around?
You're right, I interchanged heads and tails.
Where do thirders get it wrong?
In this forum and others many people prove in different ways that 50% is the good answer, but now here is my take on exactly where the thirders get trapped in their reasoning (not speaking about those who answer the wrong question)
The confusion arise if we say that in Sleeping Beauty’s universe there is no Tuesday Heads. That is the false assumption. She lives in the same universe of all the people around her, with the only difference that in this situation she is sleeping and asked no question.
Let me rephrase the problem in a way that it is easier not to get trapped in the reasoning that there is only 3 situations. Don’t tell SB that she will not be awaken on Tuesday Heads. Before being put to sleep she knows she has 25% chance to be in any one of the 4 states when she will be waken up. Now when she wakes up ask her the confidence of the coin flipped Heads. 50% she will answer, no debate here! Now tell her we are not in the Tuesday Heads state. This is a new info but that does not change the probability the coin went Heads. The only new info for SB is: « if the coin actually went Heads, then I know for sure the day is Monday. So my confidence of Monday Heads rise from 25% to 50%, my confidence for Monday Tales is still 25% and also Tuesday Tails 25%. »
The new info of not being in the Tuesday Heads state does not change the confidence the coin went Heads or Tales, it rises her confidence of being Monday Heads. With the original problem, telling in advance that she will not be waken up Tuesday Heads is just the « new » info given in advance « hey SB, today is not Tuesday Heads »
You are absolutely right that the big mistake is thinking that there is no Tue+H. There is. But then you go off into a wild rationalization for why the incorrect answer should be considered correct.
There are four possible states of the experiment. {Mon+H,Mon+T,Tue+H,Tue+T}. Each has the same probability to be the state of the experiment at a random point during it. But when SB finds herself awake, it is an observation that the state is not Tue+H. It doesn't matter what would happen in the state Tue+H, as long as it is different than "Ask her for the probability of Heads." That includes not allowing her to observe it, but you could just as well wake her and take her shopping instead of questioning her. It is trivial to conclude that the conditional probability of Heads, given the observation that the question was asked, is 1/3.
But there is another mistake you make, when you say "This is a new info but that does not *_change_* the probability....". This is a common mistake made by beginners. A conditional probability is a different measure than the unconditional probability. It is "filtered" by the information you have that eliminates some states. So it always is a "change." Sometimes it may change to the same value, but the calculation that determines the value is different.
And that is the reason halfers make their big mistake. They do not see that something has been removed from the set of possible observations, because they concentrate too much on that fact that SB knows an observation will be made.
The probability of Heads is not conditional to her observation of being awake. Giving her the info of not being in the tuesday-heads state is not a new info about the outcome of the coin. She already knew at 100% she wound wakeup on any on the 2 outcomes
@@jonathanlavoie3115 How is it not new information? Please, provide a definition of "new information" that supports this assertion. And no, it does not have anything with knowing she would be awake. It has to do with what she knows about the experiment at the moment.
Try it this way: Say that the coin you flip on Sunday has a value of 1="Heads" and 0="Tails." Get a second coin, a different kind of coin, with a value of 2="Heads" and 0="Tails." On Monday morning, place this second coin showing Heads. On Tuesday Morning, turn of over. The value of the experiment at any point in time is the sum of these two coin's values.
Now, wake her on both days. For Question #1, ask her to write down a sample space for the current value of the experiment, and to assign a probability to each possibility. Then, administer the amnesia drug and, if it is Tuesday or the first coin is Heads, the sleep drug. Allow her to see her first response, and ask Question #2: How would she update it? After she does (or does answers, administer both drugs.
The correct sample space for Q1, on either day, is {0,1,2,3}. Each of these is definitely a possibility. The correct probabilities are each 1/4, because no probability can be different than any other and there are four of them. She could even predict this on Sunday Night, raising the question of whether Q1 is needed at all.
For Q2, she knows that the value can't be 3. But the sample space she arrived at yin Q1 (or Sunday Night) still applies, as a _prior_ sample space. The fact that 3 is no longer possible is a classic example of "new information" because it eliminates a possibility from the _prior_ sample space. It can be used to update the probabilities of the remaining values to 1/3 each.
Now, how is this different? Does the information she has in Q2 of this experiment differ, in any way, from the information she has in the usual version? Doe the _prior_ sample space not apply if the same two coins are used?
The halfer's error is that they consider value=0 and value=2 to represent the same result of the experiment. It does for the people running it, at different times, but SB only knows of only one instance.
« Allow her to see her first response, and ask Question #2 » This is new information, you’re basically telling her we are Tiesday. In SB experiment, she has
NO clue if this is Monday, or this is Tuesday and being ask the same question for the second time.
@@jonathanlavoie3115 No, in the we are actually telling her that it is not Tue+H, the same information she gets in the common variation (which, actually, is not the original problem). She still knows that Mon+H, Mon+T, and Tue+T are possibilities.
So, what is your definition of "new information" that allows an update? What makes you say this doesn't exist?
Mine is that, as an answer to the first question, she says the sample space is {Mon+H, Mon+T, Tue+H, Tue+T} and that each has a *_non-zero_* probability. But after amnesia, Tue+H has *_zero_* probability. The sample space changed. This is "new information."
Or did you ignore me asking for a definition because you don't have one?
First of all, really nice vid with your explanation and sharing your thought processing! Thank you for that and very much appreciated!
Furthermore, I do agree on you regarding the possible states. There are only two states [Monday heads] OR [Monday tails AND Thuesday tails]. In the end its still only two states in total (that's also how you logical built up, stuff as a programmer). Moreover, the question and given information is incomplete if you ask me. In fact the information given by Veritasium contains already a small difference vs the explaination on Wiki. If you truely want to answer this question from the perspective of Sleeping Beauty (and not as an outsider perspective), you need to exactly know, in detail what information is kept (eg. procedure details and if this information is lost to her during the experiment). Movreover, you also need to know, if information can be collected during a wake-up-moment. Without knowing what type of information Sleepy Beauty is holding during the expirment, you cannot answer this question from her perspective (as it was ment to). This information is important because sometimes you can not proof something but can rule out all the other things, to still come to the same conclusion. Also the question can be answered from multiple disciplines, one from a purely math point of view, but also from a psychological point of view.
To give an example;
Let assum she is aware of the expirment. In otherword, she exactly knows the procedure and all the rules of the expirement (possible states) during the expirment. In that case you has the possibilty to back-track days.
However, assuming she doesn't know she is part of an experiment, than you can also argue she isn't even aware about the two sub-states at all (Monday tails and Thuesday tails). From that point she only get the information from this single question -> "What is the probability that the coin came up head?". So based on this information the can conclude that a coin has been flipped. And based on this information only, she will definitly answer the 50/50 change (excluding the minuscule change that the coin came up on its side 😅).
Anyway keep up the good work and have a blessed day!
Even from the perspective of Sleeping Beauty the probability would be 1/2 heads and 1/2 tails. Remember, she does not remember how many times she awoke so from perspective it's always the first time, so it is always Monday. But which Monday? Monday of heads, or Monday of tails? The two Mondays are NOT the same Monday.
1. Assign an outcome on the result of a flipped coin.
(Heads will have SB wake up once on Monday. Tails will have SB wake up twice on both Monday AND Tuesday.)
2. Define the coin as being fair, meaning it has a 50% chance to land heads and 50% chance to land tails.
(This is the answer to the question, "What is the probability SB woke up because of a heads coin flip?")
3. Execute the pre-decided outcome for the coin flip.
(This is Derek's video. The coin lands on tails, therefore, SB will be awoken twice on both Monday AND Tuesday, and each time she is awoken she will be asked the question "What is the probability that you woke up because of a heads coin flip?" to which her answer will be, assuming that she is intelligent enough to understand how probability works, "It is 50%." Considering that the coin landed tails she will be asked two times the same question. And both of those times, assuming she is intelligent enough to understand how probability works, she will answer the same way. Because why would she decide to answer any different way when she knows the rules of the game before she starts playing it? Why would she answer any other way when she knows the coin is fair? And why would it matter if she gives the correct answer or the wrong answer? Her answer, correct or wrong, will NOT affect the probability that she awoke as a result of either heads or tails anyway. Because it is the the coin that which determines the probability, NOT her opinion of it.)
Actually, in the original problem, the question asked to her is “What is your credence now for the proposition that the coin landed heads?"
It ends up being a problem of how you understand the question.
But the person who originally posed the problem then went on to "prove" using conditional probabilities that the correct answer is 1/3. So this was meant to be about probabilities.
(He makes a subtle mistake, though.)
@@ronald3836 Veritasium went on to prove both answers are correct answers based on how you understand the question. If you want to be correct, it’s 1/3 and if you want the probability % it’s 1/2. The question isn’t meant to be about anything rather than being paradoxical and the answer is that it depends on how you understand the question. This whole problem proves that credence isn’t strictly based on objective probability.
Ok, but what is "credence" then? If it is exactly defined mathematically via probabilities then Marco's probability analysis is the right way. If "credence" is just a subjective feeling that cannot be evaluated as "correct" or "wrong" then any answer is OK as long as it's the actual subjective feeling of the sleeper
@@charonme Adam Elga's paper says "credence" but then treats it as a probability. He "proves" that the answer is 1/3 using conditional probabilities (but makes a questionable/incorrect assumption when doing so).
@@charonme credence is how much someone believes in a statement someone else made. If the statement is made by oneself, one’s belief in his own statement is called confidence.
This is a fake paradox abd great litmus test to figure out who actually thinks and who thinks they think.
50/50 is 50/50 regardless of what day it is. SHOCKER. 😂
Suppose you are an extra participant who is also getting woken up and going back to sleep without memory.
For you though, you’ll be woken up on Monday+Tuesday for both heads and tails. You’ll also be woken up an hour earlier than SB, and get to observe whether she wakes up.
When you first wake up, I’ll assume you’d say P(heads)=50%.
Does your probability change an hour later?
Certainly if SB doesn’t wake up then P(Heads)=100%, because that’s the only possibility.
In the case SB does wake up, can you update your probability?
Indeed. Bayes' theorem provides the formula for updating your credence that the coin landed heads conditioned on the information that Sleeping Beauty was seen to be awakened. The result is P(H) = 1/3.
Exactly my view! I don't understand how this is a problem...
"if you can't go out on surface, you have to go deeper". Let's start from verifying if what SB lives doesn't influence the answer; imagine the experiment is this: "if it's head, you'll never be awakened, if it's tails, you'll be awakened on monday and then on tuesday. When you'll be awakened you won't remember if you've ever been awakened previously and you''ll answer the question: "given you are awaken (and remember the rules of the experiment), what is the probability the coin flipped on heads?". In this scenario the SB will be awakened ONLY after a tails, so how can she think there was a probability that the coin flipped on heads if in that case she wouldn't be awaken? This shows that the question isn't "what is the probability of a fair coin to flip on heads?" and that the question "what result do you think awaken you today?" (which is equal to the original question) is different, because in this question the event "Awakening" comes to play... It's true that the fair coin gives 2 results, so 2 different events, but is also true that the question doesn't focus on those 2 events (which are 2) but on these 3 events (which are 3, not 2):
- today is monday AND the coin flipped on heads;
- today is monday AND the coin flipped on tails;
- today si tuesday AND the coin flipped on tails.
All these events (these "Awakenings") have the same probability to happen BUT only 1 is consequence of the flip of the coin on heads, so the right answer to THIS question ("given you are awake...?") is 1/3.
P.S. mathematically it can be demonstrated easily, if the steps are done coherently with the laws of the probabilistic math. Also, at 7.12 the split of 50% of probability is wrong; both the events have 50%, because the second happens the 100% of the time when the first happen, which has a 50% of probability to happen, so both share the same "global" probability to happen.
P.P.S. the probability to be AWAKEN on monday after a flip on HEADS ISN'T the probability of a fair coin to flip on HEADS.
P.P.P.S. initially i thought i could simplify the problem focusing only on the probability of the fair coin to flip on Heads (50%) but i understood this was a wrong passage after a little thinking on it.
All the awakenings do not have the same probability of being the current awakening that SB is experiencing. It's 50/25/25, not 33/33/33.
@@hypersonicpiano6120 2/3 of the wakeups result from a tails coin toss, however you are confusing 66% of the total wakeups with a 66% probability of a tails coin toss! These are two very different things.
@@hypersonicpiano6120 The number of times you wake up is irrelevant to the probability of the coin toss because the coin is tossed ONLY ONCE and that is BEFORE you go to sleep. Even if you were to wake up a million times in a row you would still exist in the second of the two possible outcomes. There can ONLY BE two possible outcomes. One outcome for heads, the other for tails. And since the coin is a fair one, meaning it's a magical coin that will randomly generate a result at 50% chance then you already know the probability that you awoke because of a heads.
Heads (50% chance) {State ---> Wakes only once on Monday}
Tails (50% chance) {State ---> Wakes twice on both Monday and Tuesday}
Heads or Tails are the ONLY states. In the parenthesis is the probability of the coin landing in that state. In the squiggly brackets is the action/s taken within that state.
In computer programming you would write it thus.
If (H == TRUE) then (A == TRUE)
Else If (T == TRUE) then (B == TRUE AND C ==TRUE)
The Monday of Heads and the Monday of Tails are NOT the same Monday. They exist in different states. With an additional difference being that the Monday of Tails is ALWAYS followed by a Tuesday of Tails. Because the Tuesday of Tails exists in the same state as the Monday of Tails.
@@hypersonicpiano6120 The question is what is the probability that the coin landed on heads. In both heads and tails she will awaken on a Monday. But only in the tails will she be awoken a second time on Tuesday. There are only two states she can be in based on the conditional statement of the coin toss. The reason there can only be two states is because the random generator we are using ONLY has two possible answers, heads or tails. If we wanted six states then we would use a six sided die. But here we use a coin. When the coin lands on heads she is awoken one time. When it lands on tails she is awoken two times. The number of times she is awoken does NOT influence the probability of the result of the coin toss. Especially considering it is a fair coin whose probability is already defined before the toss is made as being 50% for heads and 50% for tails.
@@hypersonicpiano6120 if you are right then you would observe that an awakening on monday would happen only with the 75% (50+25) of the probability, since the awakening on monday has 100% of probability to happen (in both cases, with Heads or with Tails, the awakening on monday will happen) your assumption is wrong.
Each wakeup has a 1/2 chance to happen, but as they are equally likel to happen, a 1/3 chance to be the one you are experiencing. So given you wakeup, the chance is 1/3 that you happen to be wokenup while the coin toss was heads.
"Each wakeup has a 1/2 chance to happen, but as they are equally likel to happen..."
You just contradicted yourself. Each wakeup cannot have both a 1/2 and 1/3 chance. In fact they can only have a 1/2 chance. Because the wakeup of tails in Tuesday cannot exist with the tails of Monday. Therefore, all the awakenings in tails are equivalent to same exact state of tails.
@@kaizokujimbei143 No I didn't contradict myself.
Let me elebaorate on the 2 likelihoods involved in my comment:
Likelhood1: For any of the wakeups the likelihood that said wakeup will eventually happen, is 50% (because it is only determined by the fair coin toss). (Hence why the tally in the og video shows the same number for each wakeup).
Likelhood2: Because they are equally likely to happen eventually the chance your current wakeup to be any specific wakeup is also equal, because their is no way of differentiating between them and as their are 3 options said likelyhood is 1/3 (i.e 1/3 chance that your wakeup is the heads one).
Or to put it in simpler words likelyhood 1 is "the chance that X would happen", while likelyhood 2 is "the chance X had happened (given information y)"
The latter case involves a gain in information because of knowing to have woken up. This is made obvious if you replace each wakeup with a chance to be woken up.
Example:
If heads-> a 1 time 50% chance of waking up
If tails-> 2 times a 50% chance of waking up.
If you are woken in this scenario it's suddenly pretty obvious, that you gained some information about what happened, because the chances to be woken even just once is significantly higher if it was tails.
@@OzoneTheLynx The question is NOT discussing whether each wakeup will happen eventually. The question is what is the probability that any one wakeup is the result of a heads.
And the two wakeups that happen in the tails are linked to one another. They cannot happen independently from each other. Therefore, their probability is the exact same because they both happen under tails. They happen sequentially after tails. If the tails Tuesday happened then you know that a tails Monday happened. Therefore, there are ONLY two possible states. One state for heads Monday, and one state for tails Monday and Tuesday. The reason there are ONLY two states is because the random generator we are using is a coin which can either take values of either heads or tails. If we wanted to have six states then we would use a six sided die.
The number of times you are awaken in a single state is irrelevant to the probability of the coin toss. The coin toss and whether it landed on heads is the question.
To help you understand even better. Let's say in the heads you are awoken 5 times and in the tails you are awoken six times. The probability of the coin landing on heads is still 50%. Because it is a fair coin. It will land on heads 50% of the time. And it will land on tails 50% of the times. The number of times you are awoken in each scenario does NOT influence the probability of the coin toss.
@@OzoneTheLynx Why is this so difficult for you to understand?
Derek literally gave us the answer at the beginning. There is no mystery here. At least in regards to this thought experiment about probability. No. The only mystery is WHY do people keep insisting on constructing false narratives around imagined information when the answer is already staring them in the face? Why are you doing this? You are not smarter than the obvious. You are not smarter than common sense. This is simple logic. It is the same logic used in computer programming. Am I to assume you are not a programmer or that you are so stupid that you couldn't never learn how to code even if your life depended on it? I mean, dude, come on. Use your head already. Derek himself is so obviously trying to confuse you by giving you irrelevant information or by jumping from one point to another making you erroneously think that he is convincing you of something.... magical. There is no magic here. Please, focus.
@@OzoneTheLynx And I would further go on to say that Derek's video is a social experiment. Did you notice that the title of the video updates the like to dislike ratio? I think Derek is trying to prove that the opinion with the most number of people always gets bigger and bigger. He also proves that most people are easily manipulable, hence why they believe illogical and non-factual information.
You are not operating under the guise of logic and understanding. You are operating under the belief that you can somehow crack the code of the matrix. Hence the reason why Derek claimed that is people like you who believe in the nonsense about a simulated universe. We do not live in a simulation. So stop trying to pretend to be smart.
And for your information, I stopped coding computers about 7 years ago. And I never went very far to be honest. My knowledge is small. And yet I understand a basic conditional statement when I see one. You can learn too. But you have to stop whining like a child and grow up. You cannot break logic. You cannot break objective reality. Period. End of story.
I think this simplifies it:
Once she wakes up, she is asked what was the result of the flip. If she answers wrong she is killed. What is the probability she lives? 1/2.
Now if instead of that, she only gets slapped in the face for each wrong guess (after the experiment is over), what is the probability she gets slapped? 1/2, but she gets slapped twice if she guesses heads and its tails.
Conclusion: she should bet on tails, knowing she has a 50/50 chance of getting slapped and a 100% chance of getting slapped twice as many times if she goes with heads.
The phalacy here is to treat the two days provided by tails as independant, as they are bound to happen either both or not at all.
But, if you change it up again and say that she gets slapped momentarily and forgets that too, then she only gets slapped once for both bad guesses. And I think this is the most realistic interpretation of the original question, since it implies she won't remember anything that happened during the experiment and thus, her experience should be viewed seperately for each day and not accumulatively.
So no, 1/3 doesn't make sense in any scenario that doesn't assume she will somehow count her strikes at the end of the experiment. It makes no difference to her how many times she gets it right otherwise.
And if we're talking parallel worlds? I'm pretty sure those don't add up. I may have infinite lives but I can live only one.
I was a halfer but on further reflection I became a thirder. Here's a simple way I explain it:
Imagine the experiment is carried out multiple times. On average she will be woken 1.5 times.
For simplicity, let's say we run the experiment twice, therefore she is woken an average of 3 times.
On one wakening, the coin was heads; on the other two it was tails.
The probability it was heads, therefore, is one in three.
Yes, the probability that she will answer the question correctly if she says "heads" is 1/3, because she is answering the question some redundant amount of times, according to the problem.
But the question isn't asking her to be correct about whether it came up heads or tails the last time it was flipped. It's asking her what she thinks the probability is that it came up heads. If the coin is flipped tails, yes she will be answering the question twice, but that's irrelevant. It was still a single coin flip.
Don't count the number of times she answers the question, just count the coin flips. That's what demonstrates the actual results. The question's a bit vague "what should she say...?" and it leaves some room open to interpretation. But for me, I think she should give an accurate answer, not one that protects her from being wrong more often.
If the question were more specific like, "What should she answer, if she wants the record of her answers to match probabilistically to the flip result for each time she was questioned", then I would agree should should say heads is 1 in 3.
I think "being right" and "correctly identifying the probability" are really the same thing. Though I understand your objection.
And now that you mention it, since the coin is only flipped once then...1/3 doesn't make sense.
Interesting! :)
Maybe I'll go with: if the experiment is repeated multiple times then it's 1/3; but if only performed once then it's 1/2.
What are the implications for the simulation argument?
None. The simulation argument doesn't predict our existence or nonexistence (which would correspond to SB being awake or asleep). It predicts we would observe a Universe that has limited resolution -- something we have not observed.
It's plain wrong
@@rsm3t What about quanta? What about the limited speed of light? What about uncertainty? Those fit the idea of limited resolution. And btw: the simulation could also dictate that we can't observe resolution limits.
@@RealAXork It may be plain, but I don't know if it's wrong.
@@wassollderscheiss33Michael Titelbaum analyzes quantum cases in his book, "Quitting Certainties", if you are interested in that. It's too lengthy to go into here. The finite speed of light has no bearing on the problem.
This is what happens when who doesn't have the instruments to make logical thinking expresses an opinion... A great lack in comprehension of conditional probability in all the replies.
Anyway here's a hint, the big part of the mathematicians or don't give a f about this problem, or argue on what is the real question...
Agree 😊
16:25!!!
Everytime I get convinced by a halfer argument I go back to this empirical example at 7:36 and cannot reconcile the two. If I know that "I am currently woken up" and the cases of me being woken up are counted on this paper, and there are 2x many cases of me being woken up that belong to the "tails" column, then it is logical to me that my current awakening is more likely to belong under "tails" column, and so... more likely that the coin came up tails? I don't know where is the error in this line of reasoning
The error is that the number of wakeups are completely irrelevant to the probability.
Consider this: We flip a coin repeatedly. When heads comes up I give you $1. When tails comes up I give you $2. In the end two thirds of the money you get will come from the tails results. Does that mean that tails was more likely to be thrown? No, it just shows that the rewards were different for heads and tails.
So by saying the number of wakeups equals the probability of the coin you are comparing two things that are fundamentally different (an absolute value and a percentage).
@@32fw34dgasdfg I don't know, he tosses a fair coin and just fills the columns accordingly... so what's the difference between choosing one line from this table at random (probability 2/3 of choosing the one with tails) vs waking up and so also being in a state corresponding to one of these lines (not 2/3 anymore, but why)
@@marcostrat You're probably right but I have to wrap my head around this still
@@kskcp7243 Why is it so difficult? The number of times you wake up is irrelevant to the probability of the coin toss because the coin is tossed ONLY ONCE and that is BEFORE you go to sleep. Even if you were to wake up a million times in a row you would still exist in the second of the two possible outcomes. There can ONLY BE two possible outcomes. One outcome for heads, the other for tails. And since the coin is a fair one, meaning it's a magical coin that will randomly generate a result at 50% chance then you already know the probability that you awoke because of a heads.
Heads (50% chance) {State ---> Wakes only once on Monday}
Tails (50% chance) {State ---> Wakes twice on both Monday and Tuesday}
Heads or Tails are the ONLY states. In the parenthesis is the probability of the coin landing in that state. In the squiggly brackets is the action/s taken within that state.
In computer programming you would write it thus.
If (H == TRUE) then (A == TRUE)
Else If (T == TRUE) then (B == TRUE AND C ==TRUE)
The Monday of Heads and the Monday of Tails are NOT the same Monday. They exist in different states. With an additional difference being that the Monday of Tails is ALWAYS followed by a Tuesday of Tails. Because the Tuesday of Tails exists in the same state as the Monday of Tails.
@@kskcp7243 2/3 of the wakeups result from a tails coin toss, however you (and Veritassium?) are confusing 66% of the total possible wakeup states with a 66% probability! Both are percentages, but these represent two very different things as per Marco's response. The betting odds of a sporting event do not affect the chance of a particular outcome even though most of the time the betting companies are correct (correlation vs causation).
You misunderstand that what Veratasium is making isn't a mathematical argument but a philosophical one. The math is there just to confuse people. And philosophy is there, as always to make pointless assertions that sound clever.
To be specific, he views the wake up states as aposteriori "realities" in which the beauty finds herslef in and asks how one should look at these realities. The one third argumant is that because in the case of tails you have 2 realities in which you are correct, you should go with tails, while the one half argument tells you that you don't actually know what the right answer is and are just as likely to be wrong or right, regardless of how many times that may be.
Indeed the right answer depends on the question which was cleverly presented as "what should she say?"
She can recite the bible for all I care, I don't know the woman!
Great video, Derek was not logical in his initial video and misrepresented the "halfer" position, while contradicting mathematics itself to advocate for the "thirder" position. If he was unbiased and logical as you were then the answer would have been obvious from the start. The crux of this problem is that expected value (1.5 awakenings) has been incorrectly conflated with a 1/3 probability.
Sure, so why the Sleeping Beauty has to live 3 events, which are the 3 possibile awakenings, when the coin has only 2 possibile consequences? And if the SB wouldn't be awaken if the coin flips on Heads and would be awaken only after a Tails how could she answer 1/2 at the question (and read well what is the question) "given you are awake, what is the probability the coin flipped on Heads?" That's not "what is the probability of a coin to flip on Heads?" since this question doesn't include in itself the events "awakenings".
@@antog9770 all of these questions are answered in the video, please rewatch it. These 3 live events are really two events. The fact that she is awoken twice when the coin lands tails does not change the 50% chance that it will land heads, she wakes up once, and that is the end of it.
50% of the time the coin will land heads, and in 100% of those cases she wakes up on Monday (50% total probability of taking place and that being the case when SB wakes up).
50% of the time the coin will land tails, and in 100% of those cases she will be awoken on Monday, and in 100% of those cases she will also be awoken on Tuesday. Therefore when SB opens her eyes if you tell her it landed tails there is 50% chance it is Monday and 50% it is Tuesday (because both events are equally likely - guaranteed - to happen). Multiplying this by the overall chance of a tails coin flip (50%) you get 25% chance the current awakening is Monday T and 25% Tuesday T.
Therefore when SB wakes up the probability distribution that her current awakening is one of the scenarios is 50/25/25, not 33/33/33. Since both tails awakenings take place, Veritassium counts both in the table which is an incorrect way of simulatung this situation. If you simulate it correctly you will get the 50/25/25 distribution (don't just count the total number of awakenings, but the chance that the current awakening is one of the 3 scenarios).
@@hypersonicpiano6120 there's too lack of probability knowledge to continue, you should study a bit of probability first to think to have understood this problem...
@@antog9770 thanks for pointing out the errors in my reasoning bro
@@hypersonicpiano6120 i said were you fall, MH is consequence of the flip, MT is consequence of the flip and so MH+MT=1 so the probability laws are all happy, then, only when MT happens (which isn't the flip of the coin event, this is another event) surely happens TT, so it has the same probability of MH to happen, and since MT =1/2, TT =1/2 too. The sum is greater than 100% because the TT event isn't a direct consequence of the flip (there will never be a TT first of the MT event), and when you include events which are not consequences of the causes you are considering the sum can't be 100%. Differently by the coin, the SB is directly connected to all of these 3 events, in facts she can answer in 3 possibile scenarios and in 2 of these the coin flipped on Tails, so, since she has to answer the question considering all the info she can have, the right answer is 1/3 since only in 1/3 of the events she can live the coin flipped on Heads. If you understand well the question you can understand all that comes after...
In philosophy people can talk endlessly.
In math you can do the work to get to the one and only correct answer.
In math the definition of probability is "the number of times a particular event occurs divided by the total number of possible outcomes." (You can have a different definition of probability but this is the math definition.)
If you conduct this experiment 100 times with 100 different sleeping beauties, you will get 50 heads flips and 50 tails flips.
The 50 heads beauties will be asked 50 questions.
The 50 tails beauties will be asked 100 questions.
There will be A total of 150 questions asked.
Since there are 100 flips of a fair coin, there are no other numbers possible then the ones listed above.
Each of the sleeping beauties knows the above.
When each of the beauties are woken up they are asked "What do you believe is the probability the coin landed heads?"
Each of the beauties knows the coin landed heads 50 times.
Each of the beauties knows there will be 150 total questions asked.
So from the perspective of each of the beauties the probability that the coin landed heads is
50 ÷ 150 = 1/3.
You can look at this any way you want but the above is the one and only way math looks at it.
50 heads ÷ 100 flips = 1/2
That is the probability of a fair coin toss, of course. And if sleeping beauty was woken up and asked "what is the probability of a fair coin toss landing heads" she of course will answer 1/2, every time she is woken up, no matter what day it is, and no matter what the coin toss was that decided how many times she was woken her up.
But the question she is asked is "What do you believe is the probability the coin flip made after you were first put to sleep landed heads." And she knows if it landed heads she will be asked once, and if it landed tails she will be asked twice. So she know the probability of heads is half the probability of tails. Even though the question(s) she is asked nor her answer(s) does not change the probability of a fair coin toss.
I will reply in a new post titled « where do thirders get it wrong »
@@jonathanlavoie3115 Well you can reply where/how ever you want, of course. But Marcos' video "Sleeping Beauty Problem Followup" correctly and completely describes the problem, the answers, and the source of the controversy.
Seems obvious to me that the Veritasium fellow is wrongly assuming that asking sleeping beauty "what is the probability of the coin flip being heads?" is the same as if she was to be asked, instead "what is the probability that you have either, just been awakened on a Monday to be fo!lowed by another awakenig on Tuesday, or you have just been awakened on a Monday, not to be fol!owed by another awakening on Tuesday, or you have just been awakened on Tuesday?" But these questions are not the same. Am I missing something?