The problem with figuring out which probability is the right answer to the question is that the answer changes depending on what the question is concerned with. Q1: "What is the probability that the coin came up heads?" A1: 1/2 Q2: "What is the probability that you are awake now because of a heads result?" A2: 1/3 Q3: "What is the probability that you are awake now because of a tails result?" A3: 2/3 If the question is only concerned with the coin and the coin alone, then it doesn't matter how many times one is woken up as a result.
I would agree with that. The chance for both flips are the same. The question is does it matters how many time you guess wronge or correct? 1. If your goal is never guess wrong because for example you get killed if you do so the right answer is 0.5. It does not matter what you choise. As long you say the same answers all the time! 2. If on the other hand you goal is to optimize the amount of correct answers then you have to go with tails. For example if you win one doller for every time you say the correct answer. In that case this are your choices: guess tails and it was tails-> win 1'000'001$ guess tails and it was not tails -> win nothing -> average win 500'000.5$ guess headand it was tails - > win nothing guess head and it was not tails -> win 1$ -> average win 0.5$ I know what I would pick in that case :)
say you had 3 pegs arranged in an L shape. There was a 50-50 chance of knocking down the peg on the left or the peg on the right and if the peg on the left is knocked down it will hit the peg beneath it. What are the chances of the peg on the right falling?
I think that some people have touched on this slightly, but I feel inclined to write it nonetheless: Why one way of reasoning results in 1/2 and another in 1/3 is because they answer different questions. One question is "What is the probability that the coin landed heads during the experiment" which quite clearly is 1/2, while the other question is "What is the probability you have been woken up when the coin has landed heads?" which in turn is 1/3.
John Love Right, I should have wrote "What is the probability you have been woken up because the coin has landed heads?", I expressed myself unclearly :)
except it's a half - you are in one of two worlds, both with probability of 1/2 - one of the world's is the get woken once world, and the other is the get woken twice world
This video is at significant variance with the original problem - as debated by Lewis and Elga. Stated correctly, the original protocol is as follows: if the coin lands heads, beauty will be woken on Monday, interviewed and sent home; if the coin lands tails, beauty will be woken on Monday, interviewed, put back to sleep, have her memory erased of the last awakening, woken up on Tuesday, interviewed and sent home. This gives us the traditional 1/2 vs 1/3 arguments of Lewis and Elga - based on the protocol of Heads = one awakening with no memory loss, Tails = two awakenings with one memory loss. Some statements of the problem have her being put to sleep again and woken on Wednesday.However the problem is essentially unchanged so long as only one potential memory loss can occur - between Monday and Tuesday (if the coin landed tails). The video above, for both heads and tails, adds a further memory loss before a Wednesday awakening. This potentially changes the answer. For this version, if beauty awakes and is informed it's not yet Wednesday, the Lewis and Elga probabilities for heads will update to 3/7 vs 1/3 respectively. Before Wednesday is ruled out, their estimates will be 1/2 vs 2/5.
Change to "guess whether the coin was heads or tails. If you guess H and are correct you win $1 at the end of the experiment and if you guess tails and are correct you win $0.99 at the end of the experiment (total NOT per awakening). Run a simulation 10000 times and it should be clear that guessing H will win you more so it cannot be correct that the probability of H is less than 1/3 as 1/3*1 < 2/3*0.99
*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.*
It's 1/3 because if she is woken up, there is only a 1/3 chance it was during a heads flip. Considering she is woken up, you need to adjust the probability to a conditional probability to account for this. Given that she was woken up, it is more likely it was on a tails flip.
I don’t see it that way. She is awakened in both cases with equal probability, I.e. she always wakes at the end of the experiment, and she has no idea how many times she was awakened before so she has no rational basis to assume it’s anything but 50-50. At the end of the experiment, the probability that the coin is heads or tails is 50%. If you run a simulation, the last state of the experiment will have a 50% chance of heads or tails, that was determined prior to any guesses along the way. Isn’t it the final time she is woken that matters?
It's amazing how many people don't understand basic probability. There are three awakenings. The probability of any one of them is 0.5. The sum of all *independent* outcomes is 1. But the two awakenings on Tuesday are not independent: if one happens so does the other. Ergo, the combined probability of those two awakenings is 0.5, and the total probability is the expected 1. The probability of the coin landing heads is thus 0.5. Probability is one of the most difficult areas of maths to get your head around, which is why people make boner mistakes like this.
You arrogance and stupidity is amazing. I am sure you are one of the people that not get the monty hall problem correct if you do it with 1'000'000 doors. If you like to know the correct answer look at my commen above. If you are arrogant than at least try to be correct ;)
Yes, mate, I do understand the Monty Hall problem, which has nothing to do with this problem. The error in the Sleeping Beauty problem lies in the definition of mutually exclusive. In probability, two outcomes are mutually exclusive if a single repetition of the event cannot produce both outcomes. However, the statement of the problem deceives most people into applying their own definition, which is that two outcomes are mutually exclusive if Sleeping Beauty cannot experience them at the same time. Trying to force the false definition into the problem creates the supposed paradox.
The three possible outcomes are mutually exclusive during a given interview. Exactly one of the following is true upon SB being interviewed: "Today is Monday and the coin got heads", "Today is Tuesday and the coin got heads," "Today is Tuesday and the coin got tails." Thus, they are mutually exclusive and jointly exhaustive, so their probabilities add to 1. The event in question is not the experiment as a whole, but the interview. A single repetition of the interview cannot produce both outcomes "Today is Monday" and "Today is Tuesday."
If you roll a dice and argue about the probability that a particular number is showing, then the event is the dice roll and the outcomes are the numbers. If you conduct an experiment and argue about the probability of which interview is occurring then the event is the experiment and the outcomes are the interviews. One typical way that people often use to calculate the probability in this problem is to repeat the experiment 100 times and count how often each interview occurs. When you do something like this you are counting the outcomes of an event. Does being interviewed cause the day to become Monday or cause the coin to become heads? No. The coin landing heads causes the Monday/heads interview. You've got your causality back to front. I should point out that it took me about 20 years to get my head around this problem, so it's not surprising people have trouble with it. :-)
No one's having trouble with this problem but you. Cause and effect aren't relevant here. You seem to think "Today is Monday and the coin was heads," Today is Monday and the coin was tails," and "Today is Tuesday and the coin was tails" all have 50% probability, which is absurd. Or am I misunderstanding what you believe? If you were SB and the interviewer asked you the following questions: "What is the probability that today is Monday and the coin was heads", "What is the probability that today is Monday and the coin was tails," "What is the probability that today is Tuesday and the coin was tails," how do you answer those questions?
Here’s an observation I’ve made before. The intended focus of the problem is: if the coin lands Heads she’s woken once (Monday); if Tails she’s woken twice (Monday and Tuesday) with amnesia in between. Beauty is meant to consider three indistinguishable awakenings - Heads/Monday, Tails/Monday and Tails/Tuesday. Her last memory is Sunday in all of these. It matters how many times she’s given the amnesia drug in each coin outcome. In most versions of the problem, Beauty is only given the amnesia drug if the coin landed Tails, administered after the Monday awakening, not after the Tuesday one. In that scenario, with her last memory being Sunday, there would indeed be three possible awakenings for Thirders and Halfers to dispute. Either it’s 1/3 heads and 2/3 Tails or it’s 1/2 for either - depending on which position you take.
However, in the problem stated here, Beauty is given the amnesia drug after every awakening regardless of Heads or Tails. Does this matter? Yes, because she wakes up on Wednesday with the same memory loss as the other awakenings. Therefore, before Wednesday is verbally ruled out, there are five possible awakenings: Heads/Monday, Heads/Wednesday, Tails/Monday, Tails/Tuesday and Tails/Wednesday. It’s therefore relevant to ask what her answers are before and after Beauty is told whether it’s Wednesday. If she’s a traditional ‘Thirder’ she would first say its 2/5 the coin landed Heads and 3/5 Tails; following Bayes, she would update these to 1/3 and 2/3 if told it’s not Wednesday. As a a ‘Halfer’, her first answer would be 1/2 for Heads or Tails; following Bayes, she would update to 3/7 Heads and 4/7 Tails if told it’s not Wednesday. If subsequently told it’s Monday (Tuesday and Wednesday eliminated), what is her new credence? If she previously updated from 2/5 to 1/3 for Heads, she must update again to 1/2. If she previously updated from 1/2 to 3/7 for Heads, she must update to 3/5 for Heads and 2/5 for Tails. I’ve not declared a Halfer or Thirder position here but I hope my observation about the exact number of amnesia doses on each outcome makes sense.
Let's just say that we ask Sleeping beauty what does she think the result of the coin flip, and she gives the same answer always. And we do this for so many times that any anomalies are balaced out. 1. We ask her every time we wake her up, and she says heads: She's wrigth 1 out of 3 times. 2. We ask her every time and she says tails: She's wright 2 out of 3 times. 3. We ask her only on wednesdays: She's wright one half of the time.
This is a conditional probability question. What is the chance the coin landed on heads, given that you have been woken. P(A|B)=P(AandB)/P(B) In case you did not know. (That's Probability of A given B) B=You are woken therefore P(B)=3/4 A=Coin lands heads and P(AandB)=1/4 (1/4)/(3/4)=(1/4)*(4/3)=1/3=P(A|B) QED That's not odd, because she learns that it is not (Tuesday and the coin came up heads). More than enought information to change the conditional probability.
no 1/2 because the only information she got is i woke up at least once witche is a necessary result whether they had one face or the other witche means she has no information she should respond 1/2
it depends on whether she is told the day. if she is told it is a Monday, them the answer is 1/2. this is because being women up on Monday vs Tuesday are distinct events. likewise if she wakes up on Tuesday, and is told it is Tuesday, the answer is 0 that it is heads and 1 that it is tails. if she is NOT told what day it is, then it is still 1/2 because her infotainment is the same as it was on Sunday. she isn't basing her conclusion on when she woke, but only on the fairness of the coin. Thus when she woke up, if she is not told is not relevant data since she doesn't know it
Yeah, agreed. This has got to be one of the most awful philosophical 'problems' in existence, surely. I don't know who came up with this abomination, but could this be any more inelegant? Suggestion #1: leave sleeping beauty, the fictional fairy tale character out of it completely, and just get to the math. Because I cannot get past sleeping beauty and the memory wiping sedative imagery, on a purely imaginative level. out of my head enough to concentrate on whatever the actual substance of it.
Sleeping Beauty is aware of inescapable truth of the coin having only 2 sides. The probability of the coin being heads is always 1/2. What happened and what is possible are joined in an unnatural way while you're describing this problem. They aren't the same thing at all. One is prescriptive and one is descriptive. Despite what the outcome could have been (prescriptive), Sleeping Beauty is being asked to make a statement about the past (descriptive). The only possible pasts that could exist are one where the coin landed heads and one where the coin landed tails.
In principle it should not really matter what day it is. The scenario only changes with the day and if she is awake or not. Sunday she goes to sleep, coin offers 50/50 chance. Monday she's woken up with a 50/50 chance and Tuesday she is woken up by default on the result of Monday. Telling her on Sunday the odds and the results that played out should not have any influence on the reality of the real odds of the actual chance of what is going to happen. Her memory was not wiped out on Sunday. Concluding she would know that there is a 50/50 chance of her being awake, regardless of the situation or day. The difficulty come when assigning the third day a value of chance, and to ask her the odds of her being on that day. The trouble lies with Tuesday. Assigning a value to any day as an odd of her being awake, and she having to assign a value to an inconsequential day, causes an inconsequential situation/day for the mathematical side of things, because for the 2 scenarios the odds won't change; The odds will remain: Heads/ Tail = Awake or sleep / Awake awake = which is still 50/50. Just my 2 cents.
The probability of heads is always 1/2. It does not matter how many times they wake her up, or if they tell her it's Monday. You don't have to add up the probabilities to 1, since the cases don't exclude each other.
@@Noonycurt Think of it that way: If the experiment was repeated 1,000,000 times (e.g. on one million weeks), there would be about 500,000 awakenings on Heads-Monday, about 500,000 on Tails-Monday and about 500,000 on Tails-Tuesday. Which means she is in each position 1/3 of the times, which means the probability that the the coin came up heads (i.e. that she is in situation Heads-Monday) is 1/3.
@@cube2fox That is not the probability. If she always answers "heads", she'll be right on Monday. If she always answers "tails", she'll be right on Monday and Tuesday. BUT - not because the answer is more likely to be correct, but because they asked her twice in the case of tails. So the people asking the question determine the amount of correct answers - not the coin.
@@Noonycurt Not quite, the correct answer depends both on the unconditional probably of the coin and on how many times she is awaken for heads/tails. The more often she is awaken for one outcome of the coin flip, relative to the other, the more likely it is that a single awaking occures under that outcome.
@@cube2fox That is correct. It still does not change the probability. How often which answer is correct - given is a certain toin coss - depends solely on the number of times she is asked. This is not probability. It other words: The probability for the coin showing tails is not 2/3 as the video implies. I am criticizing the video, not your statement.
There is a linguistic trick in the question. From the perspective of what Sleeping Beauty knows before the experiment starts the credence is 50-50. There is no credence after the coin is flipped because then it is 100-0 in either direction weather anyone knows it or not. So it is always the wrong answer if Sleeping Beauty give other percentages than 50-50 because she knows how the experiment was implemented. But if she is asked if it is heads or tales, and she gets the same reward for being right, she should guess the one she knows would lead to her guessing more often.
Kind of reminds me of the monty hall problem. Can't you say that the probability she's in the 1,000,000 cycle at all is 50%? It doesn't make sense to me why each awakening in the 1,000,000 cycle is treated as its own individual event when they're really essentially one long event.
because from her perspective she doesn't know which event she is on. if it's the 561,001st awakening or the 19,044th awakening, she doesn't know which it is. But she's asked every such awakening. she has to "gamble" what was the coin toss result. let's say she gets $1 each time she guesses correctly. which strategy maximizes her winnings? if you were right, then it wouldn't matter what she would bet, since the chances are 50%-50% anyway - and she could just randomly say "tails" or "heads". but, you are wrong: if she bet she's on the "long" path (i.e., on the coin flip that yields more awakenings) - she would (on average) be richer. She'd have 1 million dollars with a chance of 50%, and zero dollars with a chance of 50%, or in total, $500K expected winnings.
Sleeping beauty knows that the procedure to be adopted during the experiment (that is why she knows that three awakenings are possible). Regardless of when she wakes up, the probability of heads coming up will be 1/2. The probability that she is woken up on Monday given a head = 1 The probability that she is woken up on any other day given a tail = 1/(n) Yes there are three awakenings but they aren't equally likely. I do not understand what the problem is?
***** I still do not get it. Sleeping Beauty knows all awakenings are not equally likely. It is like saying that if that the probability that dart would land inside innermost circle on a board is 1/2 because there are two possibilities.
+John Love The probability she's going to be woken on any other day given tail is 1. The probability of the coin having been heads is different if you presupose that she has woken up - wich we obviously do. It's conditional probability.
The second paradox, that is the one with the 1/3 view applied to a million awakenings, seems related to the unexpected hanging paradox. The Tuesday awakenings can't have the same value as the Monday ones because they are contingent.
1/2 because the only information she got is i woke up at least once witche is a necessary result whether they had one face or the other witche means she has no information she should respond 1/2 the problem is defined precisely but those who try to resolve it are inprecise in there reasoning
@@derpderpina1804 But there IS more information than what you had before the experiment. If you are being asked the question then you are awake and therefore you are not on Tuesday after the coin was flipped heads. I.e. there are only 3 options you can chose from (split 1/3 vs 2/3) and not 4 (split 2/4 vs 2/4).
The flaw in logic that this paradox rests on is the wrongful partitioning of scenarios. There is really only a Scenario A and B. Because she goes to sleep, is awakened, then goes to sleep and is awakened again, we're tempted to split off each wakeful state as its own scenario. There's no reason to, though. If Sleeping Beauty wakes up on Monday in scenario B, she WILL wake up on Tuesday and Wednesday to the same question, regardless of what she does. The ONLY variable is the initial coin flip. I'm not a probability expert, but I'm sure someone who knows more can verify that in probability theory, defining a distinct scenario is redundant unless it exists as a sibling to a different scenario on a tree of potential outcomes.
This sounds like a base rate fallacy problem. I mean, there are 2 probabilities here: the probably of the coin flip, and the probability of the day, and these two probabilities are mixed together. So has anyone tried to use bayesian theory to solve this problem?
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...
The probability of monday-heads should be equal to the chance of monday-tails. At the same time, the chance of it being monday-tails should be the same as the chance of it being tuesday-tails. (If tails, 50% chance of it being the first and 50% chance of it being the second awakening.) So probability of Mon-Heads = probability of Mon-Tails = probability of Tues-Tails. So the answer should be 1/3. Another way of thinking of it is to actually do the experiment a 100 times and see how often she would be correct by saying heads. Assuming that the coin would end up heads 50 of the times would result in her being awoken 150 times, 100 of which would have been in a tails scenario.
**EDIT:** I've read the other comments and I've better realized what the second implied question is that is not the actual question. The implied question is "What are the odds you've awoken on any particular day in the heads or tails timeline?". If that were the question, then it would be justified to use a matrix style probability analysis and the go assign different/equal probabilities to each day. But that isn't the scenario presented. The question is "What is the probability the coin was flipped heads". It doesn't matter which day you're asked in which timeline because you don't have any additional knowledge, so you should answer as if the coin hadn't been flipped yet, 50-50. This isn't a gambling game so the analysis which split the probability into it's subset days is incorrect. -------------------------------------------------- It seems the problem lies in the method of analysis, or in the assumptions used to conclude the matrix style analysis is suitable for this question. There are a few probabilities being calculated here and they're being confused and substituted throughout the analysis. But more importantly, I think the issue is the application of "Probability Theory" here and with the questions asked and the assumption that the different days are mutually exclusive events. Probability 1) Both timelines do have a 50% chance of occurring because as was defined, the coin has fair odds. This should be the end of this thought experiment, because it was defined. There are no other odds or true probabilities to consider here. But let's note some other things that were added. Probability 2) But then each day was split up, which is a separate probability all together. This is no longer a chance that heads or tails occurred, but instead the chance that sleeping beauty was woken up on Monday or Tuesday with the certainty that Tails was flipped. The appropriate question to reveal this probability is, "What is the chance it is Monday or Tuesday?" Splitting each day into different chances doesn't make sense here. Both day will happen because time continues on, they are casual and are not exclusive of each other. The video seems to subtly change what question that is really being asked each time the probabilities are adjusted. The problem at some points is that the video is equivocating each day, or each question as equally likely to happen, which isn't the case. The string of tails questions is 50% likely to happen, and the heads question is 50% likely to happen. Here's the appropriate way to think about it. A question is just a transaction of information. Asking a question multiple times doesn't change the probability of the event happening, which is the flaw in reasoning in this video. If someone flips a fair coin across the world and we wonder, what are the chances it was heads, wondering it a second time doesn't introduce any information about what has happened and it doesn't have any influence on what happened, it is still a 50-50 coin. If we are given information about the coin, then we can make further deductions. If we're awoken and told that it's Tuesday, then we know for certain it's Tails. If we're not, then the chances are still 50-50.
I was tempted, very tempted to agree with you, however the nature of the sampling actually does change what data pool is represented by a context-free sample. When you state: "The string of tails questions is 50% likely to happen, and the [string of] heads question[s] is 50% likely to happen" you are of course correct, but the problem is the naive sample can only account for the likelihood of it's own chance of happening. The 50/50 probability collapses because we no longer have access to it. If we're just flipping coins (forget the sleeping beauty story) and I tell you that you'll be able to sample the results, but not fairly, it makes more sense (at least to me). If the result is heads I'll let you see the result once after you report your guess. We end the experiment. If the result is tails I'll let you see the result 100 times, taking your "guess" before seeing the result each time. After each time seeing the result we'll wipe your memory and bring you back to the test area as if for the first time. For me this experiment is 9/10ths psychology and 1/10th epistemology which is why it's so interesting. I don't know what the proper terminology is but it seems we have a predisposition to hold onto the 50/50 "facts" like handrails when the method of sampling clearly puts them in a supporting role.
Biomirth Youre way over complicating this. I'll show you why. I just flipped a coin. What are the odds it's heads? You don't need to think about "sampling" or over complicate it with something "collapsing". The question is simple and your answer should be 50-50, because you have no other clues, even though I already flipped the coin that doesn't matter because you have no knowledge or evidence of what happened except the nature of the coin. Now, I'm going to ask you a second time, what are the odds I flipped heads. Given no other information except it a a fair coin, you should once again say 50-50. Me asking multiple times doesn't change the inherent odds of that coin being 50-50. If I ask you tomorrow after you get a concussion the coin is still a 50-50 coin. Without any clues to what happened you should answer with the knowledge that you have, and the only knowledge Sleeping Beauty has is the coin is fair. End of discussion. There is no need to bring in any extra analysis. It's all excessive to what Sleeping Beauty is actually asked about the knowledge Sleeping Beauty actually has. Any extra analysis is answering a different question or with information about the past she doesn't have.
"the only knowledge Sleeping Beauty has is the coin is fair. End of discussion." No. Sleeping Beauty also has knowledge of how many times she'll be sampled (sounds dirty) in either case. Please see: www.reddit.com/r/philosophy/comments/2xhaxs/epistemology_the_sleeping_beauty_problem/ and of course: en.wikipedia.org/wiki/Sleeping_Beauty_problem
Biomirth She knows how many times she could be sampled, but it doesn't feedback any information about what happened because of the sedatitive. It's not really a clue to what happened. When she's asked she isn't aware of how many times she's been asked. There are no clues to inform her of what happened. Everytime she's asked, she's asked for the first time. To here knowledge, she's only asked once though so she cannot infer anything about the coin toss that happened, she only knows for certain that the coin is fair. It's irrelevant because she doesn't know anything else. I notice on the reddit thread, the top comment also agrees with me, so I'm not sure what the point of referencing that was. I also notice the wikipedia article states for the position I'm taking "it fails", but says it fails for all the same illogical reasons in the comments here. It assumes a different question then what was actually asked. It doesn't seem anyone else notices or even acknowledges the differences in the assumed questions. The grid based analylsis is incorrect because it analyses a completely different question and scenario then what is asked. The wikipedia article uses an incorrect analysis to support the idea that the Halfer Position is false without justifying why that method of analysis in the first place. Notice in the video the person just starts splitting fractions between the days and everyone here is just accepting this is the proper analysis without justifying it.
"Everytime she's asked, she's asked for the first time. To here knowledge, she's only asked once". She's informed that in some scenarios she'll be asked more than once. Yes each time she won't know which is the case but she shouldn't simply assume "I'll only be asked once". Because of that her knowledge doesn't include a premise of "I'm only asked once". "I notice on the reddit thread, the top comment also agrees with me, so I'm not sure what the point of referencing that was." So that you would encounter other critique that may better explain the "1/3" position as well as the "1/2" position. I wasn't referencing for proof that my position is correct but to broaden the conversation. I don't see the top comment as agreeing with you though, which is odd. Funny thing is that I've come back to the "1/2" camp myself: *EDIT* : Ignore the rest because after thinking about it yet again I'm back in the 1/3'rs camp for good. In any event of waking up and being asked "what are the odds we had a heads flip or tails flip?" it is right to consider that the scenarios are uneven in terms of how likely it is you're being asked when the answer is "heads" vs "tails". *However* the probabilities are *not* shared between the heads-scenario and tails-scenario. So if there are 1000 samplings on the heads scenario and just 1 in the tails scenario there's a 50% chance you're in one of the 1000, and a 50% chance you're in the solitary-sample scenario. The chance you're in one of the 1/1000 scenarios is still only 50%. I'm sorry to say it was nothing you said that convinced me but just a reconsideration of the math. Each set of scenarios has to add up to 50%. What was missing from my analysis was that the more samples are taken in a set the less it is likely that you're in the experience of any particular one of them at the moment. Which is still odd but seems true. It's odd because for one thing if you're in the 1000-set you're *going* to experience all 1000 tests. Each of those tests is then not a fraction of a percent but 100% an event. Conversely the same is true in the 1-set. You had a 50% chance of getting here but the actual test you will receive is now an inevitability. Something about the "wholeness" of an experience (waking up and being tested) contributes to a sort of dissonance around accepting the Shroedinger's cat nature of any one of the 1-1000 tests being both "Hugely unlikely" and "1000 occurrences of a something. 1000 or none". Hard to put into words really. EDIT2: The reason for going back to the 1/3'rs camp is that even though any 1 of the 1000 tests only contributes to a total of a 50% occurrence rate, they all *do* occur if that chance occurs. You can't reduce their likelihood to 1/2000 (50% of 1/1000).
During the experiment, the rational responses are 1/3 and 1/1000002 in the two scenarios imo. Here's why: sleeping beauty actually has different knowledge of the situation before the experiment than during. Consider the case where she's not woken on Monday in the case of a heads. before the experiment she can clearly say the chance is 1/2, but during it is clearly 0, as she knows that if she gets to answer the question, the coin did not come up heads. Phrased another way: you flip a coin. while it's spinning in the air, you correctly assume it has 1/2 chance to come up heads. you see it lands tails. someone then askes "what is your creedence that the coin landed heads?" 0 is the right answer. put another way: in the original scenario, there is a 1/2 chance the coin comes up heads and a 1/2 chance it comes up tails. there is a 1/4 chance the coin comes up heads AND SB gets to comment on the probability during the experiment, and a 2/4 chance that the coin comes up tails AND she gets to comment. so, in the scenarios where she gets to comment, it came up heads 1/4 out of 3/4 of the total scenarios, or to simplify the ratio, 1/3. scanned article referenced at 4:54: i disagree with point (6). ~H2 is gained relevant information from P_ to P. I disagree with (L1) that (H1vH2vH3) is not relevant to HEADS VS TAILS. is that a typo? if (H1vH2vH3) then HEADS.
That's a completely different experiment than the first though. This is the problem with the video. It changes the implied question asked and it changes the scenario so much, of course there will be different evaluations. The problem is, the initial evaluation of the original question and scenario are wrong. It makes no sense to split the days into separate probabilities. The day doesn't matter to the question. If I flip a coin and ask you the odds it were heads, you should answer 50-50 no matter how any times I ask you. Sleeping beauty is given no clues about what happened in the past. She isn't aware how many times she was asked. Therefore, her answer should always be 50-50. Just like yours. Asking you multiple times what the odds are doesn't change the inherent fairness of the coin. The discussion should be over the instant he defined it as a fair coin. Asking about a future flip and asking about a flip you haven't seen yet and have no clues about are equivalent. It's 50-50 for the original question/scenario and it's poor form the video presented a false dilemma here.
Christopher Waldorf I apologize, as I feel as though I confused the issue by considering a similar scenario. I did so because I felt was analogous, but the probabilities were a little more intuitive, but I think the analogy was unconvincing. No prob, back to the initial scenario: she's woken on Monday on a heads and Monday and Tuesday on a tails. I think the confusion hinges on an issue of perspective. From the tester's point of view, and from Sleeping Beauty's (SB's) point of view before the experiment, the odds are indeed 1/2 that the coin comes up heads. SB's perspective during the experiment is different though. Consider this: looking at the events of Monday and Tuesday, i see 2 relevant probabilities: there's a 50:50 chance the coin comes up heads or tails, and there's a 50:50 chance that it's Monday or Tuesday. The researchers have a chance to observe all four combined possible scenarios equally. However, during the experiment, from SB's point of view, if she woke up to give her credence estimation, then Heads and Tuesday is excluded from consideration, leaving only Heads and Monday, Tails and Monday, and Tails and Tuesday. There is one scenario (generated by the fair coin) in which she gets to respond in which the coin came up heads out of a possible three, therefore, her "credence" of heads is 1/3. Let me try another analogy. Say i bet with you on a fair coin flip. You win on heads. Then, we'll flip a second special fair coin that says "Monday" on one side and "Tuesday" on the other. The second coin determines when we settle the bet. I tell you, as an additional term of the bet, if the first flip comes up heads, I'm leaving town never to contact (or pay) you again Monday night (after we would potentially meet to settle up is the second coin came up "Monday"). If it comes up tails, I'll stay in town indefinitely and the bet will be settled. Assuming we're both honest, fair players, etc. what are fair terms for this bet? They depend directly on what your chances to beat me and collect your money instead of paying me are. Well, there are 2 scenarios in which I win, and 1 in which you do, and 1 in which no one wins, so i think your chances to beat me and collect are 1/3. We bet at 2:1 odds and no one is ripped off. This scenario didn't alter the fact that the first flip was a fair coin. The day does matter to the question: we met to decide the terms and odds of the bet before we flipped the coin, and yet premise makes the day affect the fair odds. Maybe another way to think of how a coin can become a 1/3 probability generator comes from my days as a kid playing Dungeons and Dragons (yes as was/am a nerd, not at all ashamed of it). D&D uses many different sided dice: 4, 6, 8, 10, 12, and 20. Sometimes you wanted to generate a probability that didn't have a corresponding die, such as 1/3. Here's one way to do it: roll the 4 sided die, and reroll any 4 result. 4 ends up getting omitted from the possibilities, and the end result is a 1/3 probability generator. Two different coins, coin 1 and coin 2 can also substitute for a 4 sided die. Let tails on coin 1 and 2 (T1T2) equal 1, T1H2 = 2, H1T2 = 3, and H1H2 = 4, and your 2 coins became a 4 sided die substitute. Do this repeatedly and record how many coin1 flips came up heads when we omit all of the H1H2 results; it will approach 1/3 of them. That is what SB not waking up on Tuesday to report her estimation of the probability of the coin flip does: omit 1 out of 4 eqi-probable scenarios from her consideration, leaving 3 eqi-probably scenarios to be considered.
Akodo Toturi The problem is, Monday and Tuesday aren't probabilities. They don't have equal probabilities of happening. They're not mutually exclusive events. It makes nonsense to assign them seperate probabilities in the context of the first scenario. They dont effect the probability of a fair coin flipping heads in any way. The answer to that is and always is 50-50. That was defined. What happens next is purely deduction and every variant of the scenario if we use the same original question, is just a difference of deduction. For example, if I tell you "I'm going to tell you if I flip tails". And then proceed to flip the coin and say nothing. Then you can deduce what I flipped Sith certainty. If you don't get any clues as to what day it is, or what I flipped, then the only knowledge you have is you're being asked what the nature of a fair coin is. You say 50-50. If I then ask you a second time, the answer is the same. If I told you "that if I ask you a second time I definitely flipped tails", that's another scenario than the first.
the answer is 1/2 in my opinion. the explanation is over complicated but the narrator does not specifically say that sleeping beauty knows the rules of the experiment. even if she is memory wiped the only logical answer for me is 1/2. why? the experimenter ask his subject what are the chances the coin landed on heads. the coin was flipped once thus creating only two scenario that could occur regardless which day or how many times she was woken up. the consequences of the coin landing on tails is almost irrelevant at my point of view, the subject was memory wiped thus have no idea which it is. 1/3 chance could happen if the question is: what are the chances that today is tuesday?
1/2 because the only information she got is i woke up at least once witche is a necessary result whether they had one face or the other witche means she has no information she should respond 1/2 you are absolutly right
The answer to the probability question is one half, no matter how many times you wake her and ask her. A different question like; what day do you think it is would be totally different; if she always says Monday, she will be right two thirds of the time.😊
Imagine a lottery where the number of winners is decided by a coin toss. If it's heads, only one person will win, but if it's tails then there will be more than one winner. It doesn't matter how many. Assuming the probability of heads or tails is 50/50, there will be more overall winners from a tails draw than from a heads draw. If you ask a random winner if they think that heads or tails was drawn on the day they won, the most likely answer would be tails. You can adapt this to the sleeping beauty problem by having two sleeping beauties. Tell them that if the coin flips heads, one of the girls will be chosen at random to be woken, but if it's tails, they will both be woken. The coin is still 50/50, but when one of them wakes up, the chance that she is one of the two to be woken on a tails flip is double the chance that she wakes up alone on the heads flip.
The "metaphysical" odds that the fair coin will come up heads is 50% by definition, but in the final scenario, Sleeping Beauty is calculating a subjective, conditional probability. She should think, "The odds that my awakening now is occurring because the coin came up heads rather than tails is 1 in 1,000,002, given all the background information I possess." Both statements are correct. There is nothing peculiar about Sleeping Beauty's situation. (Actually, it's an exceedingly odd and disturbing scenario, but as far as probability theory goes, there is nothing strange about how to calculate the odds.) Suppose when I am not looking, someone flipped a coin, it lands in the sand, and the person picks the coin off the sand. If I turn to see the tail image imprinted on the sand, and I am asked what are the odds that the coin flip was heads, I should say "almost 100% given the evidence of the coin's imprint." By contrast, if the question is what are the odds when a fair coin is flipped that it will come up heads, the answer I should give is "50%."
Maybe because in order to go over David Lewis's reasoning would make the video at least four times as long (needing to go into Bayes' theorem and the concept of centered possible worlds), and and this is just one among many quirks that stem from analysing the Sleeping Beauty problem. Although the papers by Adam Elga (who defended the 1/3 solution) and David Lewis (who defended the 1/2 solution) were among the first, there now are over one hundred papers published in the literature (written my philosophers of science, statistical modelling experts and even theoretical physicists) about the Sleeping Beauty problem and the solution remains disputed.
If she knows it's Monday, the probability for heads is 2/3. That is correct, nothing strange there. Because we know the probability of Monday after heads is double the probability of Monday after tails(1/2=1/4 * 2), then logically knowing it is Monday means double probability of heads(2/3=1/3 * 2).
Probability is assigned based on availability of information. sleeping beauty has different information than an outside observer, so she assigns different probabilities than an observer.
Definitely. In fact, if she was told at the end of the experiment that she's been awakened both on Monday and on Tuesday - and if that were true - she'd know with 100% certainty the result of the coin flip. In this case all she knows is she's been awakened. It's not as paradoxical as it seems, I guess.
The outcome of head or tail on Tuesday is not depend on whether Sleeping Beauty is awake or sleep, or whether she knows the result of previous day In other words, it is NOT a condition for the outcome of head or tail on Tuesday. It is independent event. The assertion that Sleeping Beauty will predict the outcome of tail is 1/4 is incorrect. You are asking her to predict the outcome on Tuesday ONLY, not Monday and Tuesday are both Tail. As long as she knows it is a fair coin and she has a clear mind (not superstitious, or the god of dice is on my side), then the bet is still 0.5. We can use analogy. Let say we go to a casino and play a pair of dice against the house. We can either bet the sum of dice is even or odd. We roll first time and result is odd then we leave because we lost all our money. If the result is even, we roll one more time. Now will we bet even or odd? I'll say we can bet either even or odd, it's still 50:50 chance. The outcome does not depend on previous roll. It is also true that if I go to have a big dinner, answer a phone,...etc and leave the table after 1 roll then come back, all these things do not and will not affect the outcome of second roll. The answer is indirectly proved by the fact that casino does not change the betting return after each game no matter what you do or the outcome. If these things affect the outcome, they will sure change the return ration after each game.
It all boils down to perspective, from an external observer being a halfer is more appropriate because some forms of existence are rarer than others but from the perspective of SB a thirder is more appropriate because SB and make money by betting on the result of the coin toss
i am not confused. just go through the steps properly: the coin flip is 50/50. depending on this coin flip, we are either in the "one wakeup" path, or the "n wakeups paths". but as sleeping beauty, we would have no information about that. we cannot make any statement after being woken up that we couldn't have made before.it is either the "one wakeup path" (50%) or the other path (50% chance here, too). in case of n wakeups in a single path, to 50% of the whole path are distributed evenly to all days from sleeping beauties viewpoint. there is no paradox or weird problem.
Ok, it's a recreational puzzle, not a genuine problem. The reason it sounds paradoxical is due to conditional probabilities being unintuitive, especially if described opaquely. If the question is "what is the overall probability of heads", independently of any awakenings, then that's 1/2. If the question asked is "what is the probability of heads" given that she is awoken, then it's 1/3. I am inclined to say that if she is awoken and then informed that it is Monday, then the chance of heads should be estimated as 2/3 - however if that only discards all tests on Tuesday then it would go back to 1/2 (since there are 2 possible awakenings on Monday - one for heads and one for tails). So it depends on whether the awakenings on 'Tuesday' are taken as 'live' options or not during the questioning; if they are ruled out from the onset then there are only two options to chose from; however if they are not then there are three.
This is just another example of Bayes theorem, P(A|B)=P(A)*P(B|A)/P(B), where A is in this case that she wakes up on monday, B that the coin is heads. The probability P(B) (the probability that the coin is heads) is P(B)=P(A)*P(B|A)/P(A|B) the probability that coin is head given she wakes up on a monday (P(B|A)) is 1/2, the probability that she wakes up on a monday is 2/3 and the probability that she wakes up on a monday given that the coin is heads (P(A|B)) is 1, hence we get the same answer as in the video, 1/3.
I think the probabilities have to be different depending on which observer you are. And yes i know that this is close to QM's multiple worlds shit. However, the coin is fair and tossed in a fair way so by definition its 1/2.
It isn't crazy for her to that the Chance is 1/1000002 because is not able to remember whether she wakes up the first, the 1000002 or any day in between. So if repeat it two times and one of the coinflips is head and the others is tale, then if she assumes that the coinflip resulted in tale, then she is 1000002 times right and one time wrong.
Making up other answers to a basic question doesn't change the question or make any other answers more accurate. It doesn't matter what day it is. It doesn't matter what the coin actually said. The question isn't: What day is it? What the odds you can guess correctly? What's the probability you're correct? The question was, "what are the odds it was heads?" It's 50/50. Anybody arguing otherwise doesn't understand basic English. Even if it IS Tuesday she was STILL asked what the odds the coin was heads is. It's still 50/50. Her accuracy need not apply. Period. The probability remains the same. Period. This "paradox" is a litmus test for basic observational skill.
I think the only problems are the ones created by people not defining the probability problem precisly enough, as a result of which multiple answers are obtained.
This is simple: the odds of the coin flip being heads remains 50/50, but the odds of accurately guessing the correct result becomes 1/3. The 1/2, 1/4, 1/4 odds at 2:39 are BS. That's not how probability works. You don't split the the Tails 50/50 in half like that. It's 1/3 she guesses the day right, 1/2 she guesses the coin flip right EVERY TIME SHE IS ASKED. It's not hard.
I didn't get that either at first. That 1/2 shouldn't have been split that way OR it could have been explained better. I think they meant that Sleeping beauty has 1/2 probability of being awakened twice during the experiment and, if so happens to be, she is equally likely to have been awakened in the first day or the second one. So both awakenings happen with probability 1 if the coin lands tails (tails has of course p=1/2), but they have probability 1/2 of being the one occurred that very day.
Nicely explained. The odds of either event (the coin flip result) actually happening do not change (i.e. 50/50), but as sleeping beauty is provided more (or less) information about the event that occured in the past, her ability to precisely identify what the past event was changes. If I flipped a coin yesterday, I know what the result was and I can state with 100% accuracy what the result was. If I don't tell you what the result was, however, you have less information than I, and you can still only state with 50% accuracy what the coin flip outcome was.
The probability being 1/3 is in response to "given that you are now awake, and that there are 3 equally likely scenarios which result in you being awake, what is the probability that the coin came up heads for you to be in this situation?"
The 3 scenarios are not equally likely. P(heads, Monday) = 0.5 P(tails, Monday) = 0.25 P(tails, Tuesday) = 0.25 Proof - draw a probability tree and do the relevant multiplications.
Here's another solution. One that can't be argued with. Not that people won't try - some have, and others will - but they will be ignoring the facts when they do. What I am going to do, is change the problem so that there are four volunteers in the experiment instead of one. Those who have (you can google it), and will argue against my answer will say it is a different problem. I fully agree, it is. They will ignore that it is a 100%, completely, absolutely, no-argument-possible, identical problem for one of my volunteers. The crux is, that being a different problem allows an answer to be proven by trivial methods. And this answer applies to the one volunteer where it is a 100%, completely, absolutely, no-argument-possible, identical problem. (All the others, too, but that is a "different problem.") Call my four volunteers SB1, SB2, SB3, and SB4. Each will be wakened at least once, and maybe twice, on Monday and/or Tuesday. Based on the same coin flip. SB1 will be wakened unless the day is Monday and the coin flipped Heads. SB2 will be wakened unless the day is Monday and the coin flipped Tails. SB3 and SB4 will be wakened unless the day is Tuesday, and the coin flipped Heads, or Tails, respectively. Each knows these four schedules, but will not interact with the others while she is awake. Please note that SB3's schedule is identical to the volunteer in the original experiment. Each volunteer, when awake, will be asked for her confidence that she will be wakened exactly once in the experiment. For SB3, this is identical to the question "did the coin land Heads?" So it is SB3 who is addressing the 100%, completely, absolutely, no-argument-possible, identical problem. Say you are one of these four volunteers, and find yourself awake and asked for your confidence that you will be wakened only once. You know that three volunteers are awake at this very moment, and are addressing the same question. You know that all three of you have the same information upon which to base your confidence, but that it is true for exactly only one of the three. Your answer can only be 1/3.
+Jeff Jo If your argument were valid it also proves that the probability that the coin came up tails is 1/3. That shows that your argument is flawed, but not what the flaw is. The flaw is subtle as is often the case when someone completely changes a problem so it can be "proven by trivial methods." *"You know that all three of you have the same information upon which to base your confidence, but that it is true for exactly only one of the three."* That what is true for only one of the 3? It appears you mean that only one of the 3 will be wakened only once. That's true, but 2 of the 4 SBs will be woken only once. The 4th SB is sleeping; she's not dead. So the chance that any given SB will be awakened only once is 1/2.
Say that I am one of the four volunteers. I know that I will be awake at some point. When I am, the conditional probability that I will be wakened exactly once, given that I am awake now, is 1/3. I proved this, as you seem to agree, since the condition is that we can exclude the fourth volunteer. What you seem disagree with, as evidenced by your claim the fourth volunteer can't be ignored, is whether I am asked to provide a conditional probability or an unconditional one. This difference is why this question was filed under "Epistemology," which loosely means how our knowledge of the possibilities should be applied. The issue in the original problem, is how many outcomes SB can include in her sample space. Are there only two (Heads and Tails), one of which she observes twice but without the capability to distinguish between the observations? In this case, an awake volunteer has no information that would make conditional probability different from unconditional, and the answer is indeed 1/2. Or are there four, one of which she can't observe at all? The point of my argument is that she still has knowledge of that possible outcome when she is not existing in it, so it can - AND MUST - be included. With the knowledge that only three of the original four outcomes are consistent with her evidence, conditional probability is appropriate, and the answer is 1/3. Another comment I made here addresses this in a different way. What if, on Tuesday after Heads, SB is wakened but the interview is skipped. Instead, she is taken to Disneyworld. In this case, her answer when she is interviewed is clearly 1/3. Why should her inability to observe that day, in the original, change it to 1/2?
It is wrong to infer the probability of the outcome of the coin toss from the number of possible awakening events. The probability of the outcome being heads is always 1/2. Unless new information is added to the problem (Information Is added, but she forgets it every time). So when she asks the question "What are the chances of me being awake?" she will know that the chances are 100% for heads, and 100% for tails. Since she can't count the number of instances, for her, Heads and tails are the same thing. The 'instances of awakening' variable can not be considered by her, so it is irrelevant, and since that is the only piece of new information, the probability goes back to 50/50.
(you might be trolling, but I don't expect to see trolls here) There is no third side of the coin, but 1/3 probability doesn't mean that there should be 3 possibilities, only that the possibility of interest will only come up in 1/3 of situations where the person being asked has the same information. You could be rolling a 1001-sided die that's weighted so it comes up 1 in 1/3 of rolls and each other number gets a 2/3000 chance.
If she knows it's Monday, the probability for heads is 1/2 since both heads and tails yield the same result on Monday. Since she gets her memory wiped every time in the experiment, she does not gain any new knowledge each time she wakes up. Thus, the she would always answer 1/2.
+Randy Randalman The assumption that she does not gain any information is very very wrong. She knows she was woken. That can't happen on a tuesday after the coin came up heads. It's a problem of conditional probability.
The video keeps telling me that things sound crazy or are very strange, but I'm just not seeing it. Could someone please explain. My guess is that it might sound crazy to a person who conflates the probability of the coin coming up one side or the other with the degree of credence a person who is sufficiently mathematically proficient would have about the situation. One is a property of the coin, the other is a property of the person. I don't think a sufficiently mathematically proficient person should find it strange, but then some nonsufficiently mathematically proficient people wouldn't find it strange either and I find myself thinking I am on the cusp of being proficient.
Robert Claypool yeah, I didn't find it all that difficult either. Not because I know a lot about probability; I just put myself in Sleeping Beauty's shoes and it made intuitive sense. I think he was just building it up for dramatic purposes.
Here’s an alternate version. Call it “Extreme Sleeping Beauty with 6-sided-dice”. Imagine that you have exactly the same situation as the Extreme Sleeping Beauty scenario except that a 6-sided die is rolled and you are woken up once on Monday for outcomes 1-5 and woken up a million times in a row for the outcome 6. It seems that if you have the 1/3 view then in the 6-sided-die case Sleeping Beauty on being woken up and asked “What do you think the outcome was?” she should answer “6”. If you imagine yourself in this situation though of actually being put to sleep and then woken up and asked “What do you think the outcome was?” it seems very strange (to me at least) to think at that moment you should answer “6”, that you'd be more than 100,000x likely to be right. Imagine yourself actually giving that answer and tell me it doesn't feel a little bit "dirty" somehow.
She's not asked "What do you think the outcome was?", She's would be asked in your scenario, "What is the probability that a 6 would be rolled?" This video conflates certainty about a random event with being right about a random event.
Maybe I'm misunderstanding it but it seems to me that a lot of the literature on the subject is addressing the question of "What do you think the outcome was?" Also, never mind the video, there's already a large body of analysis on the subject... see Lewis, Elga, etc. A lot of it is phrased in terms of how she should bet on the outcome. In any case I think it's fair to ask how she should answer that question.
PublicCommerce The video already confuses what question is being asked and which is the real scenario. That's the real "problem" with the Sleeping Beauty problem as I've read. Those who disagree are either analyzing a different scenario, or analyzing the same scenario with a subtlety different question asked to Sleeping Beauty, and noone seems to care to be precise about that.
A simple way to demonstrate why the "Halfer" view is wrong is by getting money involved :) Let's say she gets $1 whenever she guesses the coin toss correctly - in each awakening. Let's also imagine we're doing the 1 million awakening version (for "Tails") What strategy should she choose? She should obviously always say "Tails", because there's a 50% chance she'd come out with $1M and 50% chance she'd come out empty handed. The "50-50" strategy (of randomly giving out an answer) would yield her only half of that, on average: 250K. So, if she wishes to be right more often and get richer, she should go with the "Thirder" approach. Now, to contrast, imagine that she were NOT paid $1 per correct guess, but instead - she were asked to provide a set of cards with "T" or "H" on them, reflecting the relative odds she believes that the coin came out tails or heads". e.g. if she thinks there's 50% she should give one "T" and one "H" cards. if she thinks there's "2/3" it came out Tails, she should give two "T" cards and one "H" card. The cards would then be tallied together, for a sum, e.g. " 1 million "T" and 1 million "H". If the coin came out Heads, only the "H" cards would be kept, and all the "T" would be tossed, and vice versa. Then this entire experiment is repeated 10,000 times, each time yielding a total tally S1, S2, S3, and so on. The proportions of total tails and heads across all experiments would be compared to the proportion of "H" and "T" cards (S1+S2+S3...), and she would get $1M if she manages to reduce the difference between the ratio of "T" and "H" to the actual proportion of coin tosses below some threshold (e.g. below 1%) What strategy she should then choose? Obviously she should contribute an equal amount of "T" and "H" cards, because that's the only strategy that would minimize the error. So - you can see here that there's a DIFFERENCE between the global proportions when viewed in aggregate - and - the proportions when viewed from the perspective of an individual awakening. She can only maximize one or the other, but not both at the same time.
The betting scenarios, even if they get the right answer, are not conclusive because they don't address how multiple opportunities to bet should be handled. Instead, change the problem slightly: wake her both days! Wake her in a red room if it is Monday, or the result was Tails. Wake her in a blue room if it is Tuesday and the result was Heads. If she wakes in a blue room, she clearly knows the result was Tails. But what is the credence for Heads if she wakes in a red room? It isn't hard to prove, or believe, that it is 1/3. But, if she is in a red room, her information is exactly the same as it is in the original problem. The answer is 1/3. The error in the "halfer" argument is that it treats Tuesday, after Heads, as if it does not happen. It does, but SB does not observe it. SB knows, with the same certainty in either the original or my version, when it is not Tuesday after Heads. It is this information that allows her to update her credence from 1/2 to 1/3. How, or even whether, she would know it is Tuesday after Heads is irrelevant.
Sorry if reposting the same answer but too many to read--you state probability of H or T based on the "first" time she was woken up. This is 1/2 (given the awake options on Monday), yet you go one to switch the context of the question from "first time" to her perspective of being woken up for the first time--which then seems to be 1/3 (from all awake options of Monday and Tuesday)...It's difficult enough to focus on the paradox but you add a degree of difficulty by needing to decipher your confusing arrangement of words...I'm out!
The thing is though, if you run this experiment a million times, you will find that Sleeping Beauty is only correct 50% of the time. If you’re ascribing to Sleeping Beauty the trait that she’s knowledgeable about conditional probabilities, why wouldn’t you also allow for her to be smart enough to know her credences aren’t determinative of the outcome. Whatever she guesses about the coin, she will only be correct 50% of the time. So why wouldn’t she say this?
Her perception of the situation and any information she has does not in any way shape or form alter the probability of the coin coming up heads or tails. It has two sides. Even chance per side. Probability is always 50% for either. Nothing changes that. The mistake being made by the 1/3 people is equating HER PROBABILITY of being right about her guess with the probability of the coin coming up heads. You can make the argument that she has a 1/3 chance of guessing correctly, but that is different from the probability of the coin being heads or tails. That is ALWAYS 1/2 or 50%. No matter what her perception, when the coin flips, it is ALWAYS a 50% chance for either. ALWAYS.
I think like a gambler. Let's use the Monte-Carlo method. Repeat the experiment many times, say 600. Or imagine 600 princesses in parallel worlds. The princess always bets $1 on heads. What should she get if she's right: $2 or $3? Say $N. About 300 times it's heads and then she gains $300N-300. About 300 times it's tails and she loses $1 TWICE each time so total loss is $600. To make it fair, 300N-300=600 so N=3. Probability=1/3.
Imagine that instead of simply saying probabilities, she had to flip 6 pennies to their respective sides. So if she believes there is a 1/2 chance it's Heads, she flips 3 heads, and 3 tails. Over both possibilities of heads/tails, she will flip 10 coins to the proper side if she uses the Thirder argument, but only 9 using the Halfer.
Sleeping Beauty needs to know that the coin was fair, i.e. Prob(tails) = Prob(heads) = 1/2, in order to compute those "weird", yet not really paradoxical, values for the probability of either result of the coin flip, so I wonder: would Sleeping Beauty - or anyone else, for that matter - ever bother to think it through? I think it would make more sense if she had to guess the result to get a prize, in which case all this comes down to the additional information she gets when she knows she is awake. And information really affects probability.
You said that the first time the beauty was woken up she was going to be asked. Then the second awakening of the tails is not going to be ask. So you are only going to ask her 2 time and once for heads and once for tails. Or you poorly worded your conditions.
None of the Tail-awakenings are independent events. They’re completely determined by the original coin flip, which has 50/50 odds of coming up heads or tails. Smells like a kind of free will vs. determinism argument. She could be awoken 1,000,000 times if Tails comes up, and all 1,000,000 times are completely determined by the original coin flip. She, and others, believe each awakening has some independent existence separated from the initial coin flip. They do not. She has no choice in the matter.
It helps to write down the problem in standard way. The confusion comes from the fact that Probability of coinflip being tales given the evidence of being awake is different than Probability of being awake given the evidence of coinflip being tales. You have to apply Bayes theorem. Probability of being awake given the con flip is tales is 1. The same goes for heads. heads = not(tales) P(tales)= 0.5 P(awake | tales) =1 P(awake | heads) = 1 P(tales | awake) = P( awake | tales)*P(tales) / P(awake) = 1*0.5/1 = 0.5 The amount of times being awake after tales or heads does not matter, because they are not independent events. Here's a different formulation of the same problem: Let's say there's a lottery where you bet on a coinflip and instantly forget what you bet on. You bet on tales. Everyone else bets on heads. What is the probability you win? obviously 1/2. What is the probability you have bet on tales, given you won? obviously 1/2. Now let's say Bob bets on tales and you and everyone else bets on heads. What is the probability you win? Obviously 1/2. What is the probability you have bet on tales, given you won? obviously 1/2.
If sleeping beauty was a betting kind of girl and was asked to wager money on which day it was this problem becomes a lot clearer in my opinion. Let's say she could place a bet each time she's woken up on whether it's Monday or Tuesday. In the 1.000.000 days scenario she would be able to bet once on monday, and 1.000.000 times on Tuesday. This means that the smart money is on betting on Tuesday. If she bets €1 each time this means she loses €1 on Monday. If it's Tuesday however she is awakend 1.000.000 times and just able to bet 1.000.000 times €1. If the odds are 50/50 this means that if she bets €1 she wins €1 which adds up to €1.000.000 For even stakes the odds therefore should be 1.000.000 to 1 against it being Monday (correct me if I'm wrong and it's actually one more or less) As is said earlier in the comments it's a question of how you phrase the problem: (by FailedNuance) Q1: "What is the probability that the coin came up heads?" A1: 1/2 Q2: "What is the probability that you are awake now because of a heads result?" A2: 1/1.000.001 Q3: "What is the probability that you are awake now because of a tails result?" A3: 1.000.000/1.000.001
Your scenario assumes that SB understands how to calculate expected values, but that whoever made the bet with her is not. Your scenario is equivalent to a bet in which she wagers on the coin toss and, if she guesses correctly, is paid €1 if the coin landed on heads and €1 million if the coin landed on tails. Of course she'll guess tails, but no one who understands probability would make that bet with her.
The video did a poor job distinguishing "credence" from "probability". If you agree with 1/2 because you think that the coin is fair then you are thinking about probability, not credence. I think that the only valid reason for agreeing with 1/2 is the "no new information gained" argument
Sleeping Beauty knows: a) there was a 50:50 chance of heads or tails; b) the rules, how things will proceed in every case; c) and she has just been woken up for the first time. Based on those three facts what probability should Sleeping Beauty assign to the coin having come up heads (or tails)? Facts b & c are irrelevant. The coin toss is always before she is woken up so waking her up doesn't change the outcome. (cavet only if heads or tails does not preclude her being woken) Being woken up offers Sleep Beauty no additional information about the coin toss since it was going to happen no matter how the toss came out. That leaves fact a so her best guess is 1/2.
+cgm778 You completely ignored the heart of the problem. She is sedated, and her memory completely erased 0:54. So when she wakes up the coin was tossed either one or two times, giving a 4 possible results, but she doesn't know. If I have a coin here with me and the last time I tossed it was heads. I may have being tossed it one time or two times, only I know. I'm telling you know, I would not write this comment if I didn't had a heads in my tosses. So, what are the chances that I tossed the coin two times? It's the same problem.
+cgm778 Only facts b and c are not irrelevant. Fact c should be she has been awoken at all anyway. From c and b follows that it can't be tuesday after the coin showed heads. This changes the probability she ought to assign.
Do philosophers know that they can specifically simulate this scenario on a computer and empirically observe the result using Monte Carlo? That will decisively settle the matter beyond any dispute, no matter how rigorous you think your logic is.
Do we need a program to conclude that a coin toss that is defined to be fair (50-50) has a 50-50 chance of flipping heads? The video above seems to be analyzing subtly different questions than the one posted/presented.
Christopher Waldorf He's trying to alter the probabilities using conditionals. It's similar to the Monty Hall problem . There are very well-defined rules for this process and how to assign probabilities given new data. So it's really odd to me that a graduate student in philosophy apparently can't solve it. This is like first-week probability theory. You just have to be very specific in the rules and the information.
AntiCitizenX I haven't studied the Monty Hall problem, but it's clear to me that the implied question changes to something other than what is originally stated which is why it seems there's some "odd contradiction". In the every scenario which Sleeping Beauty is given no specific clues as to which timelines she is in, she should answer the likelihood a coin was flipped heads is 50-50. The analysis that splits the days into separate probabilities analyses a completely different question. If the question is, "What is the probability you've been awoken on Monday or Tuesday", or if the question is, "What is the probability you've been awoken on Monday AND the coin is heads, or Monday AND the coin is tails, or Tuesday AND the coin is tails", then assigning split probabilities to each day in the analysis makes sense. But those aren't the questions that were originally asked.
As a problem in probability this is completely trivial. P(heads | interview) = 1/3, this is easily demonstrated by observing the sample space and noting that the three possible outcomes have equal probability. This can also be established intuitively by simply noting that she'll be interviewed twice as often when tails has occurred as when heads has occurred, so if she were to always guess tails, she would be right two thirds of the time, while if she were to always guess heads, she would be right only one third of the time. So I don't quite get where the confusion is coming in here? I can't stress enough how trivial this is as a probability problem. You would expect first year probability students to spend no more than a few brief moments establishing that P(heads | interview) = 1/3, let alone experts, who would mostly solve this without pause. Meanwhile P(heads | monday) = 1/2 is similarly trivial to establish. The presentation seems to suggest that these two facts are somehow contradictory, when they are clearly not. In fact, knowing one of these two conditional probabilities makes establishing the other relatively simple. The line of "If you think that sleeping beauty thinks the probability should be 1/2 when she doesn't know what day it is, I can prove to you..." seems highly redundant. We could in theory prove whatever we like from here by the principle of explosion. I'm obviously missing something fundamental here, but I can't see what that might be... Sleeping Beauty's credence should correspond to P(heads | interview), as this accounts for all of the information available to her. This can be further examined by considering the problem modified as a gambling problem. If the interviewers offer a payout of R when she correctly guesses the outcome of the flip, she would be wise to bet any amount less than 2R/3 on a tails outcome, while she could only bet up to R/3 on a heads outcome while still expecting to come out on top. The only room for debate here is about whether her credence should correspond with her confidence that she can correctly answer the interviewer's questions, or with something else. But this seems tenuous at best. To me it's completely clear that her credence should optimize her ability to correctly answer the interviewer's question, otherwise it seems like a pointless concept. Is there some reasonable justification why she should have credence 1/2? The video just seems to say "You might think that the answer is clearly 1/2", which for someone totally unfamiliar with probability is quite reasonable, but certainly no one who has studied probability would think this. You then consider the sample space to arrive at the correct answer and say "but that sounds crazy doesn't it?" No! It sounds like how probabilities work! Then you say "It seems like she doesn't learn anything new when she wakes up for the first time..." Of course, if she knew she was waking up for the first time then we'd be looking at P(heads | interviewed for the first time) = P(heads | Monday) = 0.5. Still trivial, still not contradictory - she doesn't know she's waking up for the first time in the original formulation. I'm perplexed by the simplicity of this problem... How could this warrant serious academic debate? I mean it is the kind of thing that would make a good quiz in the first week of probability 101, not the kind of thing that warrants a back and forth of published articles. I'm torn between thinking "I must be missing something here", and looking at it again and going "Nope! It's still trivial." Perhaps I'd do well to actually read some of those papers. All I see is a debate of whether to evaluate probability correctly, or to do it incorrectly. Seems like an obvious choice.
it makes sense, but that was a really odd example. sleeping beauty is a story about a girl who falls asleep once for a very long time, so why use the connection here when the person is woken up 1,000,000 times. what would anybody get out of waking a person up 1,000,000 times, how would that even work out? why not something more feasible like 5 times a day? what kind of sedative were you using? i thought there was no drug that could wipe memory; that was a myth from the coldwar, right? there were a lot of details to get caught on while imagining the experiment.
It's not meant to be a literal analysis of a scenario that could happen in our world; it's an allegory meant to present a logical problem. That's why Sleeping Beauty doesn't consider the small but non-negative probability that a meteor from space strikes the lab killing her and the researchers before the experiment is completed, or terrorist attack, or any one of many real world infringements to the premise. The point of the story is to answer the question within it's boundaries, not side-step them.
Nice explanation. A similar problem exists with regard to whether we are characters in an advanced civilization's simulation. Elon Musk thinks so. "“There’s a one in billions chance we’re in base reality” Musk said. Meaning, the life we are living is most likely a video game simulation, and there’s a one in billions chance we are in living in reality. Although not everyone will agree with Musk’s opinion surrounding the topic, the tech expert did make some valid points that leave us to ponder."
no 1/2 because the only information she got is i woke up at least once witche is a necessary result whether they had one face or the other witche means she has no information she should respond 1/2
@@derpderpina1804 3:18 "Before she was put to sleep, but AFTER she was told everything about the experiment..." Nope, we are asking her at a point in time when she is completely aware of the experimental protocol. Lemme ask you this: Imagine the experiment has been run 1000 times. What is the mean number of times the will be woken up? Answer: 1500; 500 following H tosses, 1000 following T tosses. Question 2: She is asked what she thinks the result was, every awakening exactly once and recordings of all of those askings are stored. Pulling out a random recording out of that set of 1500, which case is more probable? Answer 2: 2/3 that the random awakening is after a T toss, as opposed the 1/3 after heads. As the maker of the video said, she has enough information to transition to this objective observer statistical view. Such transition is the whole point of rationality. She is correcting for the bias incurred by the H toss being comparatively hard to observe. (Edit: Forgot to add timestamp.)
Tuesday always happens though. This is the problem with the split analysis. To sleeping beauty she is not aware of how many times she is asked or what day it is when she's asked. It's impossible for her to draw any clues about the coin that was flipped, so without any additional knowledge, she should simply just say the likelihood of coin that was fair flipping heads is 50-50, because that is the only information about the flip she has access to.
Introducing a logical error to make the problem a paradox is nonsense. The probability is always 1/2. But if you count 1 for each Heads and 2 for each Tails, and play 1 000 000 times, the probability that the "number" of Tails be higher than the number of Heads is very high. Counting two for Tails is what is done with Sleeping Beauty by waking her twice if it's Tails, giving her twice more chances to be right (2/3 instead of 1/3) by choosing Tails. In other words, if the question is what are the probabilities that Sleeping Beauty be right when choosing Heads or Tails as the outcome of the flipped coin, the answer is obviously 1/3 - 2/3 since if she is right choosing Heads, she is right once but if she is right choosing Tails, she is right twice. But that is a different question. The real question is what are the probabilities that the coin went up Heads or Tails, and the answer is and will always be 1/2. The "paradox" is based upon the confusion between the probabilities of an outcome and the probabilities to be right when predicting that outcome. If I flip a coin right on my desk and ask you what is the probability that it is now (after flipping) Heads or Tails, you should answer that the probability is either 1 or 0 for each. That is very different from the probability you be right guessing whether it's Heads or Tails, because this is an event yet to come.
The paradox always lies in how the question is asked. Heads and Tails are merely two categories of possibility. The heads group and the tails group. Once you're IN the tails group it really doesn't matter how many times you wake her up because that DECISION point is now over. The question should be what do you think are the odds that you were or will be woken up twice? Those odds are 50 / 50. Or what are the odds that you are in the 1st half of the tails experiment, 2nd half of the tails experiment or only outcome of the heads experiment??? Then the answer would be 1/3. This isn't philosophy or math it's just a riddle with hidden information and turns of phrases. I mean... it's cute, but it's not as interesting as even the game show three doors question is. That one actually deals with a little bit of mathematics and doesn't rest on twisting words to create a paradox that doesn't exist.
Wll I think she actually can't determine the probability becouse she is put in a situation where she doesn't know the conditions to determend the probability.
![PHILOSOPHY - Language: Conditionals #4 [HD]](http://i.ytimg.com/vi/W0BG49ir32E/mqdefault.jpg)
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The problem with figuring out which probability is the right answer to the question is that the answer changes depending on what the question is concerned with.
Q1: "What is the probability that the coin came up heads?"
A1: 1/2
Q2: "What is the probability that you are awake now because of a heads result?"
A2: 1/3
Q3: "What is the probability that you are awake now because of a tails result?"
A3: 2/3
If the question is only concerned with the coin and the coin alone, then it doesn't matter how many times one is woken up as a result.
I would agree with that. The chance for both flips are the same. The question is does it matters how many time you guess wronge or correct?
1. If your goal is never guess wrong because for example you get killed if you do so the right answer is 0.5. It does not matter what you choise. As long you say the same answers all the time!
2. If on the other hand you goal is to optimize the amount of correct answers then you have to go with tails. For example if you win one doller for every time you say the correct answer. In that case this are your choices:
guess tails and it was tails-> win 1'000'001$
guess tails and it was not tails -> win nothing
-> average win 500'000.5$
guess headand it was tails - > win nothing
guess head and it was not tails -> win 1$
-> average win 0.5$
I know what I would pick in that case :)
Just the same as I thought... It is not about the proberbility of a coin flip is behaving unusual.. It is all about the question no being specified :S
say you had 3 pegs arranged in an L shape. There was a 50-50 chance of knocking down the peg on the left or the peg on the right and if the peg on the left is knocked down it will hit the peg beneath it. What are the chances of the peg on the right falling?
L shape? beneath it?
can you make this a little clearer?
"The first time we wake sleeping beauty up"
Doesnt this just kill the whole riddle?
I wonder how many people have watched this after playing Zero Time Dilemma ?
+Zeptic HD that's why you're my hero ! I think it's some how significant to the game given the starting scene also involves a coin toss.
+Zeptic HD I wonder if the player is shown the "third" option at the start when they seemingly escape .
I've yet to finish the game yet though
Zero Time Dilemma
ME!!!
Ahahaha, these games are a never ending mindfuck
I think that some people have touched on this slightly, but I feel inclined to write it nonetheless:
Why one way of reasoning results in 1/2 and another in 1/3 is because they answer different questions. One question is "What is the probability that the coin landed heads during the experiment" which quite clearly is 1/2, while the other question is "What is the probability you have been woken up when the coin has landed heads?" which in turn is 1/3.
+Robert Bereza The probability that she has been woken up when the coin has landed heads is 1.
John Love Right, I should have wrote "What is the probability you have been woken up because the coin has landed heads?", I expressed myself unclearly :)
+Robert Bereza Or, if you are sleeping beauty, "What is the probability that you will sleep tomorrow?"
@@robertbereza6335 Robert I think the second question is what was asked, so the answer would be 1/3...
except it's a half - you are in one of two worlds, both with probability of 1/2 - one of the world's is the get woken once world, and the other is the get woken twice world
How to explain conditional probability without mentioning the term once. xD
This video is at significant variance with the original problem - as debated by Lewis and Elga. Stated correctly, the original protocol is as follows: if the coin lands heads, beauty will be woken on Monday, interviewed and sent home; if the coin lands tails, beauty will be woken on Monday, interviewed, put back to sleep, have her memory erased of the last awakening, woken up on Tuesday, interviewed and sent home. This gives us the traditional 1/2 vs 1/3 arguments of Lewis and Elga - based on the protocol of Heads = one awakening with no memory loss, Tails = two awakenings with one memory loss.
Some statements of the problem have her being put to sleep again and woken on Wednesday.However the problem is essentially unchanged so long as only one potential memory loss can occur - between Monday and Tuesday (if the coin landed tails).
The video above, for both heads and tails, adds a further memory loss before a Wednesday awakening. This potentially changes the answer. For this version, if beauty awakes and is informed it's not yet Wednesday, the Lewis and Elga probabilities for heads will update to 3/7 vs 1/3 respectively. Before Wednesday is ruled out, their estimates will be 1/2 vs 2/5.
Thank you kind sir. For being smart
Change to "guess whether the coin was heads or tails. If you guess H and are correct you win $1 at the end of the experiment and if you guess tails and are correct you win $0.99 at the end of the experiment (total NOT per awakening). Run a simulation 10000 times and it should be clear that guessing H will win you more so it cannot be correct that the probability of H is less than 1/3 as 1/3*1 < 2/3*0.99
Im so confused.
Goddammit, the answer is 1/2. Anything else is over thinking the problem. Oh, and BTW, the dress is blue and black.
*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.*
It's 1/3 because if she is woken up, there is only a 1/3 chance it was during a heads flip. Considering she is woken up, you need to adjust the probability to a conditional probability to account for this. Given that she was woken up, it is more likely it was on a tails flip.
I don’t see it that way. She is awakened in both cases with equal probability, I.e. she always wakes at the end of the experiment, and she has no idea how many times she was awakened before so she has no rational basis to assume it’s anything but 50-50. At the end of the experiment, the probability that the coin is heads or tails is 50%. If you run a simulation, the last state of the experiment will have a 50% chance of heads or tails, that was determined prior to any guesses along the way. Isn’t it the final time she is woken that matters?
It's amazing how many people don't understand basic probability. There are three awakenings. The probability of any one of them is 0.5. The sum of all *independent* outcomes is 1. But the two awakenings on Tuesday are not independent: if one happens so does the other. Ergo, the combined probability of those two awakenings is 0.5, and the total probability is the expected 1. The probability of the coin landing heads is thus 0.5. Probability is one of the most difficult areas of maths to get your head around, which is why people make boner mistakes like this.
You arrogance and stupidity is amazing. I am sure you are one of the people that not get the monty hall problem correct if you do it with 1'000'000 doors. If you like to know the correct answer look at my commen above. If you are arrogant than at least try to be correct ;)
Yes, mate, I do understand the Monty Hall problem, which has nothing to do with this problem.
The error in the Sleeping Beauty problem lies in the definition of mutually exclusive. In probability, two outcomes are mutually exclusive if a single repetition of the event cannot produce both outcomes. However, the statement of the problem deceives most people into applying their own definition, which is that two outcomes are mutually exclusive if Sleeping Beauty cannot experience them at the same time. Trying to force the false definition into the problem creates the supposed paradox.
The three possible outcomes are mutually exclusive during a given interview. Exactly one of the following is true upon SB being interviewed: "Today is Monday and the coin got heads", "Today is Tuesday and the coin got heads," "Today is Tuesday and the coin got tails." Thus, they are mutually exclusive and jointly exhaustive, so their probabilities add to 1. The event in question is not the experiment as a whole, but the interview. A single repetition of the interview cannot produce both outcomes "Today is Monday" and "Today is Tuesday."
If you roll a dice and argue about the probability that a particular number is showing, then the event is the dice roll and the outcomes are the numbers. If you conduct an experiment and argue about the probability of which interview is occurring then the event is the experiment and the outcomes are the interviews. One typical way that people often use to calculate the probability in this problem is to repeat the experiment 100 times and count how often each interview occurs. When you do something like this you are counting the outcomes of an event.
Does being interviewed cause the day to become Monday or cause the coin to become heads? No. The coin landing heads causes the Monday/heads interview. You've got your causality back to front.
I should point out that it took me about 20 years to get my head around this problem, so it's not surprising people have trouble with it. :-)
No one's having trouble with this problem but you. Cause and effect aren't relevant here. You seem to think "Today is Monday and the coin was heads," Today is Monday and the coin was tails," and "Today is Tuesday and the coin was tails" all have 50% probability, which is absurd. Or am I misunderstanding what you believe? If you were SB and the interviewer asked you the following questions:
"What is the probability that today is Monday and the coin was heads",
"What is the probability that today is Monday and the coin was tails,"
"What is the probability that today is Tuesday and the coin was tails,"
how do you answer those questions?
People need to stop talking about theories and arguments, relating to this problem. The answer is 1/2.
Here’s an observation I’ve made before. The intended focus of the problem is: if the coin lands Heads she’s woken once (Monday); if Tails she’s woken twice (Monday and Tuesday) with amnesia in between. Beauty is meant to consider three indistinguishable awakenings - Heads/Monday, Tails/Monday and Tails/Tuesday. Her last memory is Sunday in all of these.
It matters how many times she’s given the amnesia drug in each coin outcome. In most versions of the problem, Beauty is only given the amnesia drug if the coin landed Tails, administered after the Monday awakening, not after the Tuesday one. In that scenario, with her last memory being Sunday, there would indeed be three possible awakenings for Thirders and Halfers to dispute. Either it’s 1/3 heads and 2/3 Tails or it’s 1/2 for either - depending on which position you take.
However, in the problem stated here, Beauty is given the amnesia drug after every awakening regardless of Heads or Tails. Does this matter? Yes, because she wakes up on Wednesday with the same memory loss as the other awakenings. Therefore, before Wednesday is verbally ruled out, there are five possible awakenings: Heads/Monday, Heads/Wednesday, Tails/Monday, Tails/Tuesday and Tails/Wednesday. It’s therefore relevant to ask what her answers are before and after Beauty is told whether it’s Wednesday.
If she’s a traditional ‘Thirder’ she would first say its 2/5 the coin landed Heads and 3/5 Tails; following Bayes, she would update these to 1/3 and 2/3 if told it’s not Wednesday. As a a ‘Halfer’, her first answer would be 1/2 for Heads or Tails; following Bayes, she would update to 3/7 Heads and 4/7 Tails if told it’s not Wednesday.
If subsequently told it’s Monday (Tuesday and Wednesday eliminated), what is her new credence? If she previously updated from 2/5 to 1/3 for Heads, she must update again to 1/2. If she previously updated from 1/2 to 3/7 for Heads, she must update to 3/5 for Heads and 2/5 for Tails.
I’ve not declared a Halfer or Thirder position here but I hope my observation about the exact number of amnesia doses on each outcome makes sense.
I'm here because of hear it from "Zero Escape : Zero Time Dilemma" game
Let's just say that we ask Sleeping beauty what does she think the result of the coin flip, and she gives the same answer always. And we do this for so many times that any anomalies are balaced out.
1. We ask her every time we wake her up, and she says heads: She's wrigth 1 out of 3 times.
2. We ask her every time and she says tails: She's wright 2 out of 3 times.
3. We ask her only on wednesdays: She's wright one half of the time.
I like this content
This is a conditional probability question. What is the chance the coin landed on heads, given that you have been woken.
P(A|B)=P(AandB)/P(B) In case you did not know. (That's Probability of A given B)
B=You are woken therefore P(B)=3/4
A=Coin lands heads and P(AandB)=1/4
(1/4)/(3/4)=(1/4)*(4/3)=1/3=P(A|B) QED
That's not odd, because she learns that it is not (Tuesday and the coin came up heads). More than enought information to change the conditional probability.
P(B) = 1, because SB will always be awakened.
P(A&B) = 1/2, because SB will always be awakened after a head is flipped.
no 1/2 because the only information she got is i woke up at least once witche is a necessary result whether they had one face or the other witche means she has no information she should respond 1/2
it depends on whether she is told the day. if she is told it is a Monday, them the answer is 1/2. this is because being women up on Monday vs Tuesday are distinct events. likewise if she wakes up on Tuesday, and is told it is Tuesday, the answer is 0 that it is heads and 1 that it is tails.
if she is NOT told what day it is, then it is still 1/2 because her infotainment is the same as it was on Sunday. she isn't basing her conclusion on when she woke, but only on the fairness of the coin. Thus when she woke up, if she is not told is not relevant data since she doesn't know it
Is this a matter of perspectives where 1/2 would be from the perspective of the experimenter?
I've never been so confused.
Yeah, agreed. This has got to be one of the most awful philosophical 'problems' in existence, surely. I don't know who came up with this abomination, but could this be any more inelegant? Suggestion #1: leave sleeping beauty, the fictional fairy tale character out of it completely, and just get to the math. Because I cannot get past sleeping beauty and the memory wiping sedative imagery, on a purely imaginative level. out of my head enough to concentrate on whatever the actual substance of it.
Sleeping Beauty is aware of inescapable truth of the coin having only 2 sides. The probability of the coin being heads is always 1/2. What happened and what is possible are joined in an unnatural way while you're describing this problem.
They aren't the same thing at all. One is prescriptive and one is descriptive. Despite what the outcome could have been (prescriptive), Sleeping Beauty is being asked to make a statement about the past (descriptive). The only possible pasts that could exist are one where the coin landed heads and one where the coin landed tails.
_" what the outcome could have been (prescriptive)"._ The word you're looking for is "conditional". "Prescriptive" concerns moral commands.
In principle it should not really matter what day it is. The scenario only changes with the day and if she is awake or not. Sunday she goes to sleep, coin offers 50/50 chance. Monday she's woken up with a 50/50 chance and Tuesday she is woken up by default on the result of Monday. Telling her on Sunday the odds and the results that played out should not have any influence on the reality of the real odds of the actual chance of what is going to happen. Her memory was not wiped out on Sunday. Concluding she would know that there is a 50/50 chance of her being awake, regardless of the situation or day. The difficulty come when assigning the third day a value of chance, and to ask her the odds of her being on that day.
The trouble lies with Tuesday. Assigning a value to any day as an odd of her being awake, and she having to assign a value to an inconsequential day, causes an inconsequential situation/day for the mathematical side of things, because for the 2 scenarios the odds won't change; The odds will remain: Heads/ Tail = Awake or sleep / Awake awake = which is still 50/50. Just my 2 cents.
The probability of heads is always 1/2. It does not matter how many times they wake her up, or if they tell her it's Monday.
You don't have to add up the probabilities to 1, since the cases don't exclude each other.
However, if they ask her: "In what quadrant do you think you are?", then the probabilities would be 1/2, 1/4 and 1/4 respectively.
@@Noonycurt Think of it that way: If the experiment was repeated 1,000,000 times (e.g. on one million weeks), there would be about 500,000 awakenings on Heads-Monday, about 500,000 on Tails-Monday and about 500,000 on Tails-Tuesday. Which means she is in each position 1/3 of the times, which means the probability that the the coin came up heads (i.e. that she is in situation Heads-Monday) is 1/3.
@@cube2fox That is not the probability. If she always answers "heads", she'll be right on Monday. If she always answers "tails", she'll be right on Monday and Tuesday. BUT - not because the answer is more likely to be correct, but because they asked her twice in the case of tails. So the people asking the question determine the amount of correct answers - not the coin.
@@Noonycurt Not quite, the correct answer depends both on the unconditional probably of the coin and on how many times she is awaken for heads/tails. The more often she is awaken for one outcome of the coin flip, relative to the other, the more likely it is that a single awaking occures under that outcome.
@@cube2fox That is correct. It still does not change the probability. How often which answer is correct - given is a certain toin coss - depends solely on the number of times she is asked. This is not probability. It other words: The probability for the coin showing tails is not 2/3 as the video implies. I am criticizing the video, not your statement.
There is a linguistic trick in the question. From the perspective of what Sleeping Beauty knows before the experiment starts the credence is 50-50. There is no credence after the coin is flipped because then it is 100-0 in either direction weather anyone knows it or not. So it is always the wrong answer if Sleeping Beauty give other percentages than 50-50 because she knows how the experiment was implemented.
But if she is asked if it is heads or tales, and she gets the same reward for being right, she should guess the one she knows would lead to her guessing more often.
Kind of reminds me of the monty hall problem. Can't you say that the probability she's in the 1,000,000 cycle at all is 50%? It doesn't make sense to me why each awakening in the 1,000,000 cycle is treated as its own individual event when they're really essentially one long event.
because from her perspective she doesn't know which event she is on. if it's the 561,001st awakening or the 19,044th awakening, she doesn't know which it is. But she's asked every such awakening.
she has to "gamble" what was the coin toss result.
let's say she gets $1 each time she guesses correctly.
which strategy maximizes her winnings?
if you were right, then it wouldn't matter what she would bet, since the chances are 50%-50% anyway - and she could just randomly say "tails" or "heads".
but, you are wrong: if she bet she's on the "long" path (i.e., on the coin flip that yields more awakenings) - she would (on average) be richer. She'd have 1 million dollars with a chance of 50%, and zero dollars with a chance of 50%, or in total, $500K expected winnings.
Sleeping beauty knows that the procedure to be adopted during the experiment (that is why she knows that three awakenings are possible).
Regardless of when she wakes up, the probability of heads coming up will be 1/2.
The probability that she is woken up on Monday given a head = 1
The probability that she is woken up on any other day given a tail = 1/(n)
Yes there are three awakenings but they aren't equally likely.
I do not understand what the problem is?
*****
I still do not get it. Sleeping Beauty knows all awakenings are not equally likely.
It is like saying that if that the probability that dart would land inside innermost circle on a board is 1/2 because there are two possibilities.
+John Love The probability she's going to be woken on any other day given tail is 1.
The probability of the coin having been heads is different if you presupose that she has woken up - wich we obviously do. It's conditional probability.
The second paradox, that is the one with the 1/3 view applied to a million awakenings, seems related to the unexpected hanging paradox. The Tuesday awakenings can't have the same value as the Monday ones because they are contingent.
Many of these so called 'problems' suffer from not being defined precisely
1/2 because the only information she got is i woke up at least once witche is a necessary result whether they had one face or the other witche means she has no information she should respond 1/2 the problem is defined precisely but those who try to resolve it are inprecise in there reasoning
Rather they suffer simply by many NOT able to think right lol 1/2 Obvious
@@derpderpina1804 But there IS more information than what you had before the experiment. If you are being asked the question then you are awake and therefore you are not on Tuesday after the coin was flipped heads. I.e. there are only 3 options you can chose from (split 1/3 vs 2/3) and not 4 (split 2/4 vs 2/4).
The flaw in logic that this paradox rests on is the wrongful partitioning of scenarios. There is really only a Scenario A and B.
Because she goes to sleep, is awakened, then goes to sleep and is awakened again, we're tempted to split off each wakeful state as its own scenario. There's no reason to, though. If Sleeping Beauty wakes up on Monday in scenario B, she WILL wake up on Tuesday and Wednesday to the same question, regardless of what she does. The ONLY variable is the initial coin flip.
I'm not a probability expert, but I'm sure someone who knows more can verify that in probability theory, defining a distinct scenario is redundant unless it exists as a sibling to a different scenario on a tree of potential outcomes.
This sounds like a base rate fallacy problem. I mean, there are 2 probabilities here: the probably of the coin flip, and the probability of the day, and these two probabilities are mixed together. So has anyone tried to use bayesian theory to solve this problem?
Can you state the problem again without moving through these slides? So disturbing
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...
The probability of monday-heads should be equal to the chance of monday-tails. At the same time, the chance of it being monday-tails should be the same as the chance of it being tuesday-tails. (If tails, 50% chance of it being the first and 50% chance of it being the second awakening.) So probability of Mon-Heads = probability of Mon-Tails = probability of Tues-Tails. So the answer should be 1/3.
Another way of thinking of it is to actually do the experiment a 100 times and see how often she would be correct by saying heads. Assuming that the coin would end up heads 50 of the times would result in her being awoken 150 times, 100 of which would have been in a tails scenario.
Confusing... because the approach is flawed in so many ways. Its 1/2 in all scenarios.
**EDIT:** I've read the other comments and I've better realized what the second implied question is that is not the actual question. The implied question is "What are the odds you've awoken on any particular day in the heads or tails timeline?". If that were the question, then it would be justified to use a matrix style probability analysis and the go assign different/equal probabilities to each day. But that isn't the scenario presented. The question is "What is the probability the coin was flipped heads". It doesn't matter which day you're asked in which timeline because you don't have any additional knowledge, so you should answer as if the coin hadn't been flipped yet, 50-50. This isn't a gambling game so the analysis which split the probability into it's subset days is incorrect.
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It seems the problem lies in the method of analysis, or in the assumptions used to conclude the matrix style analysis is suitable for this question. There are a few probabilities being calculated here and they're being confused and substituted throughout the analysis. But more importantly, I think the issue is the application of "Probability Theory" here and with the questions asked and the assumption that the different days are mutually exclusive events.
Probability 1) Both timelines do have a 50% chance of occurring because as was defined, the coin has fair odds. This should be the end of this thought experiment, because it was defined. There are no other odds or true probabilities to consider here. But let's note some other things that were added.
Probability 2) But then each day was split up, which is a separate probability all together. This is no longer a chance that heads or tails occurred, but instead the chance that sleeping beauty was woken up on Monday or Tuesday with the certainty that Tails was flipped. The appropriate question to reveal this probability is, "What is the chance it is Monday or Tuesday?"
Splitting each day into different chances doesn't make sense here. Both day will happen because time continues on, they are casual and are not exclusive of each other.
The video seems to subtly change what question that is really being asked each time the probabilities are adjusted. The problem at some points is that the video is equivocating each day, or each question as equally likely to happen, which isn't the case.
The string of tails questions is 50% likely to happen, and the heads question is 50% likely to happen.
Here's the appropriate way to think about it. A question is just a transaction of information. Asking a question multiple times doesn't change the probability of the event happening, which is the flaw in reasoning in this video. If someone flips a fair coin across the world and we wonder, what are the chances it was heads, wondering it a second time doesn't introduce any information about what has happened and it doesn't have any influence on what happened, it is still a 50-50 coin. If we are given information about the coin, then we can make further deductions. If we're awoken and told that it's Tuesday, then we know for certain it's Tails. If we're not, then the chances are still 50-50.
I was tempted, very tempted to agree with you, however the nature of the sampling actually does change what data pool is represented by a context-free sample.
When you state: "The string of tails questions is 50% likely to happen, and the [string of] heads question[s] is 50% likely to happen" you are of course correct, but the problem is the naive sample can only account for the likelihood of it's own chance of happening. The 50/50 probability collapses because we no longer have access to it.
If we're just flipping coins (forget the sleeping beauty story) and I tell you that you'll be able to sample the results, but not fairly, it makes more sense (at least to me). If the result is heads I'll let you see the result once after you report your guess. We end the experiment. If the result is tails I'll let you see the result 100 times, taking your "guess" before seeing the result each time. After each time seeing the result we'll wipe your memory and bring you back to the test area as if for the first time.
For me this experiment is 9/10ths psychology and 1/10th epistemology which is why it's so interesting. I don't know what the proper terminology is but it seems we have a predisposition to hold onto the 50/50 "facts" like handrails when the method of sampling clearly puts them in a supporting role.
Biomirth Youre way over complicating this. I'll show you why. I just flipped a coin. What are the odds it's heads? You don't need to think about "sampling" or over complicate it with something "collapsing". The question is simple and your answer should be 50-50, because you have no other clues, even though I already flipped the coin that doesn't matter because you have no knowledge or evidence of what happened except the nature of the coin. Now, I'm going to ask you a second time, what are the odds I flipped heads. Given no other information except it a a fair coin, you should once again say 50-50. Me asking multiple times doesn't change the inherent odds of that coin being 50-50. If I ask you tomorrow after you get a concussion the coin is still a 50-50 coin. Without any clues to what happened you should answer with the knowledge that you have, and the only knowledge Sleeping Beauty has is the coin is fair. End of discussion. There is no need to bring in any extra analysis. It's all excessive to what Sleeping Beauty is actually asked about the knowledge Sleeping Beauty actually has. Any extra analysis is answering a different question or with information about the past she doesn't have.
"the only knowledge Sleeping Beauty has is the coin is fair. End of discussion."
No. Sleeping Beauty also has knowledge of how many times she'll be sampled (sounds dirty) in either case.
Please see: www.reddit.com/r/philosophy/comments/2xhaxs/epistemology_the_sleeping_beauty_problem/
and of course:
en.wikipedia.org/wiki/Sleeping_Beauty_problem
Biomirth She knows how many times she could be sampled, but it doesn't feedback any information about what happened because of the sedatitive. It's not really a clue to what happened. When she's asked she isn't aware of how many times she's been asked. There are no clues to inform her of what happened. Everytime she's asked, she's asked for the first time. To here knowledge, she's only asked once though so she cannot infer anything about the coin toss that happened, she only knows for certain that the coin is fair. It's irrelevant because she doesn't know anything else.
I notice on the reddit thread, the top comment also agrees with me, so I'm not sure what the point of referencing that was.
I also notice the wikipedia article states for the position I'm taking "it fails", but says it fails for all the same illogical reasons in the comments here. It assumes a different question then what was actually asked. It doesn't seem anyone else notices or even acknowledges the differences in the assumed questions. The grid based analylsis is incorrect because it analyses a completely different question and scenario then what is asked. The wikipedia article uses an incorrect analysis to support the idea that the Halfer Position is false without justifying why that method of analysis in the first place. Notice in the video the person just starts splitting fractions between the days and everyone here is just accepting this is the proper analysis without justifying it.
"Everytime she's asked, she's asked for the first time. To here knowledge, she's only asked once".
She's informed that in some scenarios she'll be asked more than once. Yes each time she won't know which is the case but she shouldn't simply assume "I'll only be asked once". Because of that her knowledge doesn't include a premise of "I'm only asked once".
"I notice on the reddit thread, the top comment also agrees with me, so I'm not sure what the point of referencing that was."
So that you would encounter other critique that may better explain the "1/3" position as well as the "1/2" position. I wasn't referencing for proof that my position is correct but to broaden the conversation. I don't see the top comment as agreeing with you though, which is odd.
Funny thing is that I've come back to the "1/2" camp myself:
*EDIT* : Ignore the rest because after thinking about it yet again I'm back in the 1/3'rs camp for good.
In any event of waking up and being asked "what are the odds we had a heads flip or tails flip?" it is right to consider that the scenarios are uneven in terms of how likely it is you're being asked when the answer is "heads" vs "tails". *However* the probabilities are *not* shared between the heads-scenario and tails-scenario. So if there are 1000 samplings on the heads scenario and just 1 in the tails scenario there's a 50% chance you're in one of the 1000, and a 50% chance you're in the solitary-sample scenario. The chance you're in one of the 1/1000 scenarios is still only 50%.
I'm sorry to say it was nothing you said that convinced me but just a reconsideration of the math. Each set of scenarios has to add up to 50%. What was missing from my analysis was that the more samples are taken in a set the less it is likely that you're in the experience of any particular one of them at the moment. Which is still odd but seems true. It's odd because for one thing if you're in the 1000-set you're *going* to experience all 1000 tests. Each of those tests is then not a fraction of a percent but 100% an event. Conversely the same is true in the 1-set. You had a 50% chance of getting here but the actual test you will receive is now an inevitability. Something about the "wholeness" of an experience (waking up and being tested) contributes to a sort of dissonance around accepting the Shroedinger's cat nature of any one of the 1-1000 tests being both "Hugely unlikely" and "1000 occurrences of a something. 1000 or none".
Hard to put into words really.
EDIT2: The reason for going back to the 1/3'rs camp is that even though any 1 of the 1000 tests only contributes to a total of a 50% occurrence rate, they all *do* occur if that chance occurs. You can't reduce their likelihood to 1/2000 (50% of 1/1000).
During the experiment, the rational responses are 1/3 and 1/1000002 in the two scenarios imo. Here's why: sleeping beauty actually has different knowledge of the situation before the experiment than during. Consider the case where she's not woken on Monday in the case of a heads. before the experiment she can clearly say the chance is 1/2, but during it is clearly 0, as she knows that if she gets to answer the question, the coin did not come up heads.
Phrased another way: you flip a coin. while it's spinning in the air, you correctly assume it has 1/2 chance to come up heads. you see it lands tails. someone then askes "what is your creedence that the coin landed heads?" 0 is the right answer.
put another way: in the original scenario, there is a 1/2 chance the coin comes up heads and a 1/2 chance it comes up tails. there is a 1/4 chance the coin comes up heads AND SB gets to comment on the probability during the experiment, and a 2/4 chance that the coin comes up tails AND she gets to comment. so, in the scenarios where she gets to comment, it came up heads 1/4 out of 3/4 of the total scenarios, or to simplify the ratio, 1/3.
scanned article referenced at 4:54: i disagree with point (6). ~H2 is gained relevant information from P_ to P. I disagree with (L1) that (H1vH2vH3) is not relevant to HEADS VS TAILS. is that a typo? if (H1vH2vH3) then HEADS.
That's a completely different experiment than the first though. This is the problem with the video. It changes the implied question asked and it changes the scenario so much, of course there will be different evaluations. The problem is, the initial evaluation of the original question and scenario are wrong. It makes no sense to split the days into separate probabilities. The day doesn't matter to the question. If I flip a coin and ask you the odds it were heads, you should answer 50-50 no matter how any times I ask you. Sleeping beauty is given no clues about what happened in the past. She isn't aware how many times she was asked. Therefore, her answer should always be 50-50. Just like yours. Asking you multiple times what the odds are doesn't change the inherent fairness of the coin. The discussion should be over the instant he defined it as a fair coin. Asking about a future flip and asking about a flip you haven't seen yet and have no clues about are equivalent. It's 50-50 for the original question/scenario and it's poor form the video presented a false dilemma here.
Christopher Waldorf I apologize, as I feel as though I confused the issue by considering a similar scenario. I did so because I felt was analogous, but the probabilities were a little more intuitive, but I think the analogy was unconvincing. No prob, back to the initial scenario: she's woken on Monday on a heads and Monday and Tuesday on a tails.
I think the confusion hinges on an issue of perspective. From the tester's point of view, and from Sleeping Beauty's (SB's) point of view before the experiment, the odds are indeed 1/2 that the coin comes up heads. SB's perspective during the experiment is different though. Consider this: looking at the events of Monday and Tuesday, i see 2 relevant probabilities: there's a 50:50 chance the coin comes up heads or tails, and there's a 50:50 chance that it's Monday or Tuesday. The researchers have a chance to observe all four combined possible scenarios equally. However, during the experiment, from SB's point of view, if she woke up to give her credence estimation, then Heads and Tuesday is excluded from consideration, leaving only Heads and Monday, Tails and Monday, and Tails and Tuesday. There is one scenario (generated by the fair coin) in which she gets to respond in which the coin came up heads out of a possible three, therefore, her "credence" of heads is 1/3.
Let me try another analogy. Say i bet with you on a fair coin flip. You win on heads. Then, we'll flip a second special fair coin that says "Monday" on one side and "Tuesday" on the other. The second coin determines when we settle the bet. I tell you, as an additional term of the bet, if the first flip comes up heads, I'm leaving town never to contact (or pay) you again Monday night (after we would potentially meet to settle up is the second coin came up "Monday"). If it comes up tails, I'll stay in town indefinitely and the bet will be settled. Assuming we're both honest, fair players, etc. what are fair terms for this bet? They depend directly on what your chances to beat me and collect your money instead of paying me are. Well, there are 2 scenarios in which I win, and 1 in which you do, and 1 in which no one wins, so i think your chances to beat me and collect are 1/3. We bet at 2:1 odds and no one is ripped off. This scenario didn't alter the fact that the first flip was a fair coin.
The day does matter to the question: we met to decide the terms and odds of the bet before we flipped the coin, and yet premise makes the day affect the fair odds.
Maybe another way to think of how a coin can become a 1/3 probability generator comes from my days as a kid playing Dungeons and Dragons (yes as was/am a nerd, not at all ashamed of it). D&D uses many different sided dice: 4, 6, 8, 10, 12, and 20. Sometimes you wanted to generate a probability that didn't have a corresponding die, such as 1/3. Here's one way to do it: roll the 4 sided die, and reroll any 4 result. 4 ends up getting omitted from the possibilities, and the end result is a 1/3 probability generator. Two different coins, coin 1 and coin 2 can also substitute for a 4 sided die. Let tails on coin 1 and 2 (T1T2) equal 1, T1H2 = 2, H1T2 = 3, and H1H2 = 4, and your 2 coins became a 4 sided die substitute. Do this repeatedly and record how many coin1 flips came up heads when we omit all of the H1H2 results; it will approach 1/3 of them.
That is what SB not waking up on Tuesday to report her estimation of the probability of the coin flip does: omit 1 out of 4 eqi-probable scenarios from her consideration, leaving 3 eqi-probably scenarios to be considered.
Akodo Toturi The problem is, Monday and Tuesday aren't probabilities. They don't have equal probabilities of happening. They're not mutually exclusive events. It makes nonsense to assign them seperate probabilities in the context of the first scenario. They dont effect the probability of a fair coin flipping heads in any way. The answer to that is and always is 50-50. That was defined. What happens next is purely deduction and every variant of the scenario if we use the same original question, is just a difference of deduction. For example, if I tell you "I'm going to tell you if I flip tails". And then proceed to flip the coin and say nothing. Then you can deduce what I flipped Sith certainty. If you don't get any clues as to what day it is, or what I flipped, then the only knowledge you have is you're being asked what the nature of a fair coin is. You say 50-50. If I then ask you a second time, the answer is the same. If I told you "that if I ask you a second time I definitely flipped tails", that's another scenario than the first.
the answer is 1/2 in my opinion. the explanation is over complicated but the narrator does not specifically say that sleeping beauty knows the rules of the experiment. even if she is memory wiped the only logical answer for me is 1/2.
why? the experimenter ask his subject what are the chances the coin landed on heads. the coin was flipped once thus creating only two scenario that could occur regardless which day or how many times she was woken up. the consequences of the coin landing on tails is almost irrelevant at my point of view, the subject was memory wiped thus have no idea which it is.
1/3 chance could happen if the question is: what are the chances that today is tuesday?
1/4 actually
Isn't it always 1/2 unless given the information about the experiment?
1/2 because the only information she got is i woke up at least once witche is a necessary result whether they had one face or the other witche means she has no information she should respond 1/2 you are absolutly right
For some reason this is still a really popular puzzle on youtube 8 years later
Bad idea watching this before going to sleep.
'Confused? Well let me make you more confused.'
Yeah. No.
I was confused, not by an apparent paradox but by a bad presentation with distracting animations. And I'm a mathematician.
The answer to the probability question is one half, no matter how many times you wake her and ask her. A different question like; what day do you think it is would be totally different; if she always says Monday, she will be right two thirds of the time.😊
Imagine a lottery where the number of winners is decided by a coin toss. If it's heads, only one person will win, but if it's tails then there will be more than one winner. It doesn't matter how many. Assuming the probability of heads or tails is 50/50, there will be more overall winners from a tails draw than from a heads draw. If you ask a random winner if they think that heads or tails was drawn on the day they won, the most likely answer would be tails.
You can adapt this to the sleeping beauty problem by having two sleeping beauties. Tell them that if the coin flips heads, one of the girls will be chosen at random to be woken, but if it's tails, they will both be woken. The coin is still 50/50, but when one of them wakes up, the chance that she is one of the two to be woken on a tails flip is double the chance that she wakes up alone on the heads flip.
The "metaphysical" odds that the fair coin will come up heads is 50% by definition, but in the final scenario, Sleeping Beauty is calculating a subjective, conditional probability. She should think, "The odds that my awakening now is occurring because the coin came up heads rather than tails is 1 in 1,000,002, given all the background information I possess." Both statements are correct.
There is nothing peculiar about Sleeping Beauty's situation. (Actually, it's an exceedingly odd and disturbing scenario, but as far as probability theory goes, there is nothing strange about how to calculate the odds.) Suppose when I am not looking, someone flipped a coin, it lands in the sand, and the person picks the coin off the sand. If I turn to see the tail image imprinted on the sand, and I am asked what are the odds that the coin flip was heads, I should say "almost 100% given the evidence of the coin's imprint." By contrast, if the question is what are the odds when a fair coin is flipped that it will come up heads, the answer I should give is "50%."
4:40 : " I can prove to you...", but you don't, you say go somewhere else and read about it. So why put this video up?
Maybe because in order to go over David Lewis's reasoning would make the video at least four times as long (needing to go into Bayes' theorem and the concept of centered possible worlds), and and this is just one among many quirks that stem from analysing the Sleeping Beauty problem. Although the papers by Adam Elga (who defended the 1/3 solution) and David Lewis (who defended the 1/2 solution) were among the first, there now are over one hundred papers published in the literature (written my philosophers of science, statistical modelling experts and even theoretical physicists) about the Sleeping Beauty problem and the solution remains disputed.
If she knows it's Monday, the probability for heads is 2/3. That is correct, nothing strange there. Because we know the probability of Monday after heads is double the probability of Monday after tails(1/2=1/4 * 2), then logically knowing it is Monday means double probability of heads(2/3=1/3 * 2).
Probability is assigned based on availability of information. sleeping beauty has different information than an outside observer, so she assigns different probabilities than an observer.
Definitely. In fact, if she was told at the end of the experiment that she's been awakened both on Monday and on Tuesday - and if that were true - she'd know with 100% certainty the result of the coin flip. In this case all she knows is she's been awakened. It's not as paradoxical as it seems, I guess.
The outcome of head or tail on Tuesday is not depend on whether Sleeping Beauty is awake or sleep, or whether she knows the result of previous day In other words, it is NOT a condition for the outcome of head or tail on Tuesday. It is independent event. The assertion that Sleeping Beauty will predict the outcome of tail is 1/4 is incorrect. You are asking her to predict the outcome on Tuesday ONLY, not Monday and Tuesday are both Tail. As long as she knows it is a fair coin and she has a clear mind (not superstitious, or the god of dice is on my side), then the bet is still 0.5.
We can use analogy. Let say we go to a casino and play a pair of dice against the house. We can either bet the sum of dice is even or odd. We roll first time and result is odd then we leave because we lost all our money. If the result is even, we roll one more time. Now will we bet even or odd? I'll say we can bet either even or odd, it's still 50:50 chance. The outcome does not depend on previous roll. It is also true that if I go to have a big dinner, answer a phone,...etc and leave the table after 1 roll then come back, all these things do not and will not affect the outcome of second roll.
The answer is indirectly proved by the fact that casino does not change the betting return after each game no matter what you do or the outcome. If these things affect the outcome, they will sure change the return ration after each game.
It all boils down to perspective, from an external observer being a halfer is more appropriate because some forms of existence are rarer than others but from the perspective of SB a thirder is more appropriate because SB and make money by betting on the result of the coin toss
i am not confused. just go through the steps properly:
the coin flip is 50/50.
depending on this coin flip, we are either in the "one wakeup" path, or the "n wakeups paths". but as sleeping beauty, we would have no information about that. we cannot make any statement after being woken up that we couldn't have made before.it is either the "one wakeup path" (50%) or the other path (50% chance here, too). in case of n wakeups in a single path, to 50% of the whole path are distributed evenly to all days from sleeping beauties viewpoint.
there is no paradox or weird problem.
Ok, it's a recreational puzzle, not a genuine problem. The reason it sounds paradoxical is due to conditional probabilities being unintuitive, especially if described opaquely.
If the question is "what is the overall probability of heads", independently of any awakenings, then that's 1/2.
If the question asked is "what is the probability of heads" given that she is awoken, then it's 1/3.
I am inclined to say that if she is awoken and then informed that it is Monday, then the chance of heads should be estimated as 2/3 - however if that only discards all tests on Tuesday then it would go back to 1/2 (since there are 2 possible awakenings on Monday - one for heads and one for tails). So it depends on whether the awakenings on 'Tuesday' are taken as 'live' options or not during the questioning; if they are ruled out from the onset then there are only two options to chose from; however if they are not then there are three.
This is just another example of Bayes theorem, P(A|B)=P(A)*P(B|A)/P(B), where A is in this case that she wakes up on monday, B that the coin is heads. The probability P(B) (the probability that the coin is heads) is P(B)=P(A)*P(B|A)/P(A|B) the probability that coin is head given she wakes up on a monday (P(B|A)) is 1/2, the probability that she wakes up on a monday is 2/3 and the probability that she wakes up on a monday given that the coin is heads (P(A|B)) is 1, hence we get the same answer as in the video, 1/3.
I think the probabilities have to be different depending on which observer you are. And yes i know that this is close to QM's multiple worlds shit. However, the coin is fair and tossed in a fair way so by definition its 1/2.
It isn't crazy for her to that the Chance is 1/1000002 because is not able to remember whether she wakes up the first, the 1000002 or any day in between. So if repeat it two times and one of the coinflips is head and the others is tale, then if she assumes that the coinflip resulted in tale, then she is 1000002 times right and one time wrong.
Making up other answers to a basic question doesn't change the question or make any other answers more accurate.
It doesn't matter what day it is.
It doesn't matter what the coin actually said.
The question isn't:
What day is it?
What the odds you can guess correctly?
What's the probability you're correct?
The question was, "what are the odds it was heads?"
It's 50/50. Anybody arguing otherwise doesn't understand basic English.
Even if it IS Tuesday she was STILL asked what the odds the coin was heads is. It's still 50/50. Her accuracy need not apply. Period.
The probability remains the same. Period. This "paradox" is a litmus test for basic observational skill.
I think the only problems are the ones created by people not defining the probability problem precisly enough, as a result of which multiple answers are obtained.
This is simple: the odds of the coin flip being heads remains 50/50, but the odds of accurately guessing the correct result becomes 1/3.
The 1/2, 1/4, 1/4 odds at 2:39 are BS. That's not how probability works. You don't split the the Tails 50/50 in half like that. It's 1/3 she guesses the day right, 1/2 she guesses the coin flip right EVERY TIME SHE IS ASKED. It's not hard.
I didn't get that either at first. That 1/2 shouldn't have been split that way OR it could have been explained better. I think they meant that Sleeping beauty has 1/2 probability of being awakened twice during the experiment and, if so happens to be, she is equally likely to have been awakened in the first day or the second one. So both awakenings happen with probability 1 if the coin lands tails (tails has of course p=1/2), but they have probability 1/2 of being the one occurred that very day.
Nicely explained. The odds of either event (the coin flip result) actually happening do not change (i.e. 50/50), but as sleeping beauty is provided more (or less) information about the event that occured in the past, her ability to precisely identify what the past event was changes. If I flipped a coin yesterday, I know what the result was and I can state with 100% accuracy what the result was. If I don't tell you what the result was, however, you have less information than I, and you can still only state with 50% accuracy what the coin flip outcome was.
The probability being 1/3 is in response to "given that you are now awake, and that there are 3 equally likely scenarios which result in you being awake, what is the probability that the coin came up heads for you to be in this situation?"
The 3 scenarios are not equally likely.
P(heads, Monday) = 0.5
P(tails, Monday) = 0.25
P(tails, Tuesday) = 0.25
Proof - draw a probability tree and do the relevant multiplications.
I think 1/3 answer for the question: "what is the probability Beauty being awakened after heads?"
Here's another solution. One that can't be argued with. Not that people won't try - some have, and others will - but they will be ignoring the facts when they do.
What I am going to do, is change the problem so that there are four volunteers in the experiment instead of one. Those who have (you can google it), and will argue against my answer will say it is a different problem. I fully agree, it is. They will ignore that it is a 100%, completely, absolutely, no-argument-possible, identical problem for one of my volunteers.
The crux is, that being a different problem allows an answer to be proven by trivial methods. And this answer applies to the one volunteer where it is a 100%, completely, absolutely, no-argument-possible, identical problem. (All the others, too, but that is a "different problem.")
Call my four volunteers SB1, SB2, SB3, and SB4. Each will be wakened at least once, and maybe twice, on Monday and/or Tuesday. Based on the same coin flip. SB1 will be wakened unless the day is Monday and the coin flipped Heads. SB2 will be wakened unless the day is Monday and the coin flipped Tails. SB3 and SB4 will be wakened unless the day is Tuesday, and the coin flipped Heads, or Tails, respectively. Each knows these four schedules, but will not interact with the others while she is awake.
Please note that SB3's schedule is identical to the volunteer in the original experiment.
Each volunteer, when awake, will be asked for her confidence that she will be wakened exactly once in the experiment. For SB3, this is identical to the question "did the coin land Heads?" So it is SB3 who is addressing the 100%, completely, absolutely, no-argument-possible, identical problem.
Say you are one of these four volunteers, and find yourself awake and asked for your confidence that you will be wakened only once. You know that three volunteers are awake at this very moment, and are addressing the same question. You know that all three of you have the same information upon which to base your confidence, but that it is true for exactly only one of the three.
Your answer can only be 1/3.
+Jeff Jo If your argument were valid it also proves that the probability that the coin came up tails is 1/3. That shows that your argument is flawed, but not what the flaw is.
The flaw is subtle as is often the case when someone completely changes a problem so it can be "proven by trivial methods."
*"You know that all three of you have the same information upon which to base your confidence, but that it is true for exactly only one of the three."*
That what is true for only one of the 3? It appears you mean that only one of the 3 will be wakened only once. That's true, but 2 of the 4 SBs will be woken only once. The 4th SB is sleeping; she's not dead. So the chance that any given SB will be awakened only once is 1/2.
Say that I am one of the four volunteers. I know that I will be awake at some point. When I am, the conditional probability that I will be wakened exactly once, given that I am awake now, is 1/3. I proved this, as you seem to agree, since the condition is that we can exclude the fourth volunteer.
What you seem disagree with, as evidenced by your claim the fourth volunteer can't be ignored, is whether I am asked to provide a conditional probability or an unconditional one. This difference is why this question was filed under "Epistemology," which loosely means how our knowledge of the possibilities should be applied.
The issue in the original problem, is how many outcomes SB can include in her sample space. Are there only two (Heads and Tails), one of which she observes twice but without the capability to distinguish between the observations? In this case, an awake volunteer has no information that would make conditional probability different from unconditional, and the answer is indeed 1/2.
Or are there four, one of which she can't observe at all? The point of my argument is that she still has knowledge of that possible outcome when she is not existing in it, so it can - AND MUST - be included. With the knowledge that only three of the original four outcomes are consistent with her evidence, conditional probability is appropriate, and the answer is 1/3.
Another comment I made here addresses this in a different way. What if, on Tuesday after Heads, SB is wakened but the interview is skipped. Instead, she is taken to Disneyworld. In this case, her answer when she is interviewed is clearly 1/3. Why should her inability to observe that day, in the original, change it to 1/2?
Very clear video! Thanks!
It is wrong to infer the probability of the outcome of the coin toss from the number of possible awakening events.
The probability of the outcome being heads is always 1/2. Unless new information is added to the problem (Information Is added, but she forgets it every time). So when she asks the question "What are the chances of me being awake?" she will know that the chances are 100% for heads, and 100% for tails. Since she can't count the number of instances, for her, Heads and tails are the same thing. The 'instances of awakening' variable can not be considered by her, so it is irrelevant, and since that is the only piece of new information, the probability goes back to 50/50.
I'm lost you only have 2 possibilities where are they getting the 3 side of the coin?
(you might be trolling, but I don't expect to see trolls here) There is no third side of the coin, but 1/3 probability doesn't mean that there should be 3 possibilities, only that the possibility of interest will only come up in 1/3 of situations where the person being asked has the same information. You could be rolling a 1001-sided die that's weighted so it comes up 1 in 1/3 of rolls and each other number gets a 2/3000 chance.
Did Arnold Zuboff (the originater of this question) just not pay attention in statistics?
If she knows it's Monday, the probability for heads is 1/2 since both heads and tails yield the same result on Monday. Since she gets her memory wiped every time in the experiment, she does not gain any new knowledge each time she wakes up. Thus, the she would always answer 1/2.
+Randy Randalman The assumption that she does not gain any information is very very wrong. She knows she was woken. That can't happen on a tuesday after the coin came up heads.
It's a problem of conditional probability.
The video keeps telling me that things sound crazy or are very strange, but I'm just not seeing it. Could someone please explain. My guess is that it might sound crazy to a person who conflates the probability of the coin coming up one side or the other with the degree of credence a person who is sufficiently mathematically proficient would have about the situation. One is a property of the coin, the other is a property of the person. I don't think a sufficiently mathematically proficient person should find it strange, but then some nonsufficiently mathematically proficient people wouldn't find it strange either and I find myself thinking I am on the cusp of being proficient.
Robert Claypool yeah, I didn't find it all that difficult either. Not because I know a lot about probability; I just put myself in Sleeping Beauty's shoes and it made intuitive sense. I think he was just building it up for dramatic purposes.
Here’s an alternate version. Call it “Extreme Sleeping Beauty with 6-sided-dice”. Imagine that you have exactly the same situation as the Extreme Sleeping Beauty scenario except that a 6-sided die is rolled and you are woken up once on Monday for outcomes 1-5 and woken up a million times in a row for the outcome 6. It seems that if you have the 1/3 view then in the 6-sided-die case Sleeping Beauty on being woken up and asked “What do you think the outcome was?” she should answer “6”. If you imagine yourself in this situation though of actually being put to sleep and then woken up and asked “What do you think the outcome was?” it seems very strange (to me at least) to think at that moment you should answer “6”, that you'd be more than 100,000x likely to be right. Imagine yourself actually giving that answer and tell me it doesn't feel a little bit "dirty" somehow.
She's not asked "What do you think the outcome was?", She's would be asked in your scenario, "What is the probability that a 6 would be rolled?" This video conflates certainty about a random event with being right about a random event.
Maybe I'm misunderstanding it but it seems to me that a lot of the literature on the subject is addressing the question of "What do you think the outcome was?" Also, never mind the video, there's already a large body of analysis on the subject... see Lewis, Elga, etc. A lot of it is phrased in terms of how she should bet on the outcome. In any case I think it's fair to ask how she should answer that question.
PublicCommerce The video already confuses what question is being asked and which is the real scenario. That's the real "problem" with the Sleeping Beauty problem as I've read. Those who disagree are either analyzing a different scenario, or analyzing the same scenario with a subtlety different question asked to Sleeping Beauty, and noone seems to care to be precise about that.
A simple way to demonstrate why the "Halfer" view is wrong is by getting money involved :)
Let's say she gets $1 whenever she guesses the coin toss correctly - in each awakening.
Let's also imagine we're doing the 1 million awakening version (for "Tails")
What strategy should she choose? She should obviously always say "Tails", because there's a 50% chance she'd come out with $1M and 50% chance she'd come out empty handed.
The "50-50" strategy (of randomly giving out an answer) would yield her only half of that, on average: 250K.
So, if she wishes to be right more often and get richer, she should go with the "Thirder" approach.
Now, to contrast, imagine that she were NOT paid $1 per correct guess, but instead - she were asked to provide a set of cards with "T" or "H" on them, reflecting the relative odds she believes that the coin came out tails or heads". e.g. if she thinks there's 50% she should give one "T" and one "H" cards. if she thinks there's "2/3" it came out Tails, she should give two "T" cards and one "H" card.
The cards would then be tallied together, for a sum, e.g. " 1 million "T" and 1 million "H". If the coin came out Heads, only the "H" cards would be kept, and all the "T" would be tossed, and vice versa.
Then this entire experiment is repeated 10,000 times, each time yielding a total tally S1, S2, S3, and so on.
The proportions of total tails and heads across all experiments would be compared to the proportion of "H" and "T" cards (S1+S2+S3...), and she would get $1M if she manages to reduce the difference between the ratio of "T" and "H" to the actual proportion of coin tosses below some threshold (e.g. below 1%)
What strategy she should then choose?
Obviously she should contribute an equal amount of "T" and "H" cards, because that's the only strategy that would minimize the error.
So - you can see here that there's a DIFFERENCE between the global proportions when viewed in aggregate - and - the proportions when viewed from the perspective of an individual awakening.
She can only maximize one or the other, but not both at the same time.
The probability is very high that Sleeping Beauty would be asleep the whole time
The betting scenarios, even if they get the right answer, are not conclusive because they don't address how multiple opportunities to bet should be handled. Instead, change the problem slightly: wake her both days! Wake her in a red room if it is Monday, or the result was Tails. Wake her in a blue room if it is Tuesday and the result was Heads.
If she wakes in a blue room, she clearly knows the result was Tails. But what is the credence for Heads if she wakes in a red room? It isn't hard to prove, or believe, that it is 1/3. But, if she is in a red room, her information is exactly the same as it is in the original problem. The answer is 1/3.
The error in the "halfer" argument is that it treats Tuesday, after Heads, as if it does not happen. It does, but SB does not observe it. SB knows, with the same certainty in either the original or my version, when it is not Tuesday after Heads. It is this information that allows her to update her credence from 1/2 to 1/3. How, or even whether, she would know it is Tuesday after Heads is irrelevant.
1 am is not the space to be thinking about this video
Sorry if reposting the same answer but too many to read--you state probability of H or T based on the "first" time she was woken up. This is 1/2 (given the awake options on Monday), yet you go one to switch the context of the question from "first time" to her perspective of being woken up for the first time--which then seems to be 1/3 (from all awake options of Monday and Tuesday)...It's difficult enough to focus on the paradox but you add a degree of difficulty by needing to decipher your confusing arrangement of words...I'm out!
The thing is though, if you run this experiment a million times, you will find that Sleeping Beauty is only correct 50% of the time. If you’re ascribing to Sleeping Beauty the trait that she’s knowledgeable about conditional probabilities, why wouldn’t you also allow for her to be smart enough to know her credences aren’t determinative of the outcome. Whatever she guesses about the coin, she will only be correct 50% of the time. So why wouldn’t she say this?
Her perception of the situation and any information she has does not in any way shape or form alter the probability of the coin coming up heads or tails.
It has two sides. Even chance per side. Probability is always 50% for either. Nothing changes that.
The mistake being made by the 1/3 people is equating HER PROBABILITY of being right about her guess with the probability of the coin coming up heads.
You can make the argument that she has a 1/3 chance of guessing correctly, but that is different from the probability of the coin being heads or tails. That is ALWAYS 1/2 or 50%.
No matter what her perception, when the coin flips, it is ALWAYS a 50% chance for either. ALWAYS.
I think like a gambler. Let's use the Monte-Carlo method. Repeat the experiment many times, say 600. Or imagine 600 princesses in parallel worlds. The princess always bets $1 on heads. What should she get if she's right: $2 or $3? Say $N.
About 300 times it's heads and then she gains $300N-300. About 300 times it's tails and she loses $1 TWICE each time so total loss is $600. To make it fair, 300N-300=600 so N=3. Probability=1/3.
Imagine that instead of simply saying probabilities, she had to flip 6 pennies to their respective sides. So if she believes there is a 1/2 chance it's Heads, she flips 3 heads, and 3 tails. Over both possibilities of heads/tails, she will flip 10 coins to the proper side if she uses the Thirder argument, but only 9 using the Halfer.
Sleeping Beauty needs to know that the coin was fair, i.e. Prob(tails) = Prob(heads) = 1/2, in order to compute those "weird", yet not really paradoxical, values for the probability of either result of the coin flip, so I wonder: would Sleeping Beauty - or anyone else, for that matter - ever bother to think it through? I think it would make more sense if she had to guess the result to get a prize, in which case all this comes down to the additional information she gets when she knows she is awake. And information really affects probability.
The answer is ask the right question, and remember to phrase it correctly
You said that the first time the beauty was woken up she was going to be asked. Then the second awakening of the tails is not going to be ask. So you are only going to ask her 2 time and once for heads and once for tails. Or you poorly worded your conditions.
None of the Tail-awakenings are independent events. They’re completely determined by the original coin flip, which has 50/50 odds of coming up heads or tails. Smells like a kind of free will vs. determinism argument. She could be awoken 1,000,000 times if Tails comes up, and all 1,000,000 times are completely determined by the original coin flip. She, and others, believe each awakening has some independent existence separated from the initial coin flip. They do not. She has no choice in the matter.
It helps to write down the problem in standard way. The confusion comes from the fact that Probability of coinflip being tales given the evidence of being awake is different than Probability of being awake given the evidence of coinflip being tales. You have to apply Bayes theorem.
Probability of being awake given the con flip is tales is 1. The same goes for heads.
heads = not(tales)
P(tales)= 0.5
P(awake | tales) =1
P(awake | heads) = 1
P(tales | awake) = P( awake | tales)*P(tales) / P(awake) = 1*0.5/1 = 0.5
The amount of times being awake after tales or heads does not matter, because they are not independent events. Here's a different formulation of the same problem:
Let's say there's a lottery where you bet on a coinflip and instantly forget what you bet on. You bet on tales. Everyone else bets on heads. What is the probability you win? obviously 1/2. What is the probability you have bet on tales, given you won? obviously 1/2. Now let's say Bob bets on tales and you and everyone else bets on heads. What is the probability you win? Obviously 1/2. What is the probability you have bet on tales, given you won? obviously 1/2.
If sleeping beauty was a betting kind of girl and was asked to wager money on which day it was this problem becomes a lot clearer in my opinion.
Let's say she could place a bet each time she's woken up on whether it's Monday or Tuesday. In the 1.000.000 days scenario she would be able to bet once on monday, and 1.000.000 times on Tuesday. This means that the smart money is on betting on Tuesday.
If she bets €1 each time this means she loses €1 on Monday.
If it's Tuesday however she is awakend 1.000.000 times and just able to bet 1.000.000 times €1.
If the odds are 50/50 this means that if she bets €1 she wins €1 which adds up to €1.000.000
For even stakes the odds therefore should be 1.000.000 to 1 against it being Monday (correct me if I'm wrong and it's actually one more or less)
As is said earlier in the comments it's a question of how you phrase the problem:
(by FailedNuance)
Q1: "What is the probability that the coin came up heads?"
A1: 1/2
Q2: "What is the probability that you are awake now because of a heads result?"
A2: 1/1.000.001
Q3: "What is the probability that you are awake now because of a tails result?"
A3: 1.000.000/1.000.001
Your scenario assumes that SB understands how to calculate expected values, but that whoever made the bet with her is not. Your scenario is equivalent to a bet in which she wagers on the coin toss and, if she guesses correctly, is paid €1 if the coin landed on heads and €1 million if the coin landed on tails. Of course she'll guess tails, but no one who understands probability would make that bet with her.
The video did a poor job distinguishing "credence" from "probability". If you agree with 1/2 because you think that the coin is fair then you are thinking about probability, not credence. I think that the only valid reason for agreeing with 1/2 is the "no new information gained" argument
+Xiaoman Chu "poor job distinguishing", heh. It's actually "made no attempt distinguishing"
Sleeping Beauty knows:
a) there was a 50:50 chance of heads or tails;
b) the rules, how things will proceed in every case;
c) and she has just been woken up for the first time.
Based on those three facts what probability should Sleeping Beauty assign to the coin having come up heads (or tails)?
Facts b & c are irrelevant.
The coin toss is always before she is woken up so waking her up doesn't change the outcome. (cavet only if heads or tails does not preclude her being woken)
Being woken up offers Sleep Beauty no additional information about the coin toss since it was going to happen no matter how the toss came out.
That leaves fact a so her best guess is 1/2.
+cgm778 You completely ignored the heart of the problem.
She is sedated, and her memory completely erased 0:54. So when she wakes up the coin was tossed either one or two times, giving a 4 possible results, but she doesn't know.
If I have a coin here with me and the last time I tossed it was heads. I may have being tossed it one time or two times, only I know. I'm telling you know, I would not write this comment if I didn't had a heads in my tosses. So, what are the chances that I tossed the coin two times? It's the same problem.
+cgm778 Only facts b and c are not irrelevant. Fact c should be she has been awoken at all anyway. From c and b follows that it can't be tuesday after the coin showed heads. This changes the probability she ought to assign.
Do philosophers know that they can specifically simulate this scenario on a computer and empirically observe the result using Monte Carlo? That will decisively settle the matter beyond any dispute, no matter how rigorous you think your logic is.
But isn't that excessive to the question that is asked?
Christopher Waldorf
It would take me less than an hour to write and run this program.
Do we need a program to conclude that a coin toss that is defined to be fair (50-50) has a 50-50 chance of flipping heads? The video above seems to be analyzing subtly different questions than the one posted/presented.
Christopher Waldorf
He's trying to alter the probabilities using conditionals. It's similar to the Monty Hall problem . There are very well-defined rules for this process and how to assign probabilities given new data. So it's really odd to me that a graduate student in philosophy apparently can't solve it. This is like first-week probability theory. You just have to be very specific in the rules and the information.
AntiCitizenX I haven't studied the Monty Hall problem, but it's clear to me that the implied question changes to something other than what is originally stated which is why it seems there's some "odd contradiction". In the every scenario which Sleeping Beauty is given no specific clues as to which timelines she is in, she should answer the likelihood a coin was flipped heads is 50-50. The analysis that splits the days into separate probabilities analyses a completely different question. If the question is, "What is the probability you've been awoken on Monday or Tuesday", or if the question is, "What is the probability you've been awoken on Monday AND the coin is heads, or Monday AND the coin is tails, or Tuesday AND the coin is tails", then assigning split probabilities to each day in the analysis makes sense. But those aren't the questions that were originally asked.
As a problem in probability this is completely trivial. P(heads | interview) = 1/3, this is easily demonstrated by observing the sample space and noting that the three possible outcomes have equal probability. This can also be established intuitively by simply noting that she'll be interviewed twice as often when tails has occurred as when heads has occurred, so if she were to always guess tails, she would be right two thirds of the time, while if she were to always guess heads, she would be right only one third of the time.
So I don't quite get where the confusion is coming in here? I can't stress enough how trivial this is as a probability problem. You would expect first year probability students to spend no more than a few brief moments establishing that P(heads | interview) = 1/3, let alone experts, who would mostly solve this without pause.
Meanwhile P(heads | monday) = 1/2 is similarly trivial to establish. The presentation seems to suggest that these two facts are somehow contradictory, when they are clearly not. In fact, knowing one of these two conditional probabilities makes establishing the other relatively simple.
The line of "If you think that sleeping beauty thinks the probability should be 1/2 when she doesn't know what day it is, I can prove to you..." seems highly redundant. We could in theory prove whatever we like from here by the principle of explosion.
I'm obviously missing something fundamental here, but I can't see what that might be... Sleeping Beauty's credence should correspond to P(heads | interview), as this accounts for all of the information available to her. This can be further examined by considering the problem modified as a gambling problem. If the interviewers offer a payout of R when she correctly guesses the outcome of the flip, she would be wise to bet any amount less than 2R/3 on a tails outcome, while she could only bet up to R/3 on a heads outcome while still expecting to come out on top.
The only room for debate here is about whether her credence should correspond with her confidence that she can correctly answer the interviewer's questions, or with something else. But this seems tenuous at best. To me it's completely clear that her credence should optimize her ability to correctly answer the interviewer's question, otherwise it seems like a pointless concept.
Is there some reasonable justification why she should have credence 1/2? The video just seems to say
"You might think that the answer is clearly 1/2",
which for someone totally unfamiliar with probability is quite reasonable, but certainly no one who has studied probability would think this. You then consider the sample space to arrive at the correct answer and say
"but that sounds crazy doesn't it?"
No! It sounds like how probabilities work! Then you say
"It seems like she doesn't learn anything new when she wakes up for the first time..."
Of course, if she knew she was waking up for the first time then we'd be looking at P(heads | interviewed for the first time) = P(heads | Monday) = 0.5. Still trivial, still not contradictory - she doesn't know she's waking up for the first time in the original formulation.
I'm perplexed by the simplicity of this problem... How could this warrant serious academic debate? I mean it is the kind of thing that would make a good quiz in the first week of probability 101, not the kind of thing that warrants a back and forth of published articles. I'm torn between thinking "I must be missing something here", and looking at it again and going "Nope! It's still trivial." Perhaps I'd do well to actually read some of those papers. All I see is a debate of whether to evaluate probability correctly, or to do it incorrectly. Seems like an obvious choice.
it makes sense, but that was a really odd example. sleeping beauty is a story about a girl who falls asleep once for a very long time, so why use the connection here when the person is woken up 1,000,000 times. what would anybody get out of waking a person up 1,000,000 times, how would that even work out? why not something more feasible like 5 times a day? what kind of sedative were you using? i thought there was no drug that could wipe memory; that was a myth from the coldwar, right? there were a lot of details to get caught on while imagining the experiment.
The "details" you got caught on were all in your head and not a part of the video.
so why call it sleeping beauty? why not just show the table?
paul netzel Just so you have something to get tripped up on.
It's not meant to be a literal analysis of a scenario that could happen in our world; it's an allegory meant to present a logical problem. That's why Sleeping Beauty doesn't consider the small but non-negative probability that a meteor from space strikes the lab killing her and the researchers before the experiment is completed, or terrorist attack, or any one of many real world infringements to the premise. The point of the story is to answer the question within it's boundaries, not side-step them.
Seems like it depends on the exact question that you are asking.
fair coin 은 휘어지지 않은 동전이 아니라 독립사건의 확률이 1/2 이 되는 동전이라는 뜻입니다. 수정 바랍니다.
Wake her up a million times on Tuesday? But that means waking her up 11 or 12 times every second! That's crazy!
Seems like the video is conflating various porbabilities together
½. She doesn't know if it is Monday or Tuesday. The only information she has to go on is her knowledge that the scenario was decided by a coin flip.
Nice explanation. A similar problem exists with regard to whether we are characters in an advanced civilization's simulation. Elon Musk thinks so.
"“There’s a one in billions chance we’re in base reality” Musk said. Meaning, the life we are living is most likely a video game simulation, and there’s a one in billions chance we are in living in reality. Although not everyone will agree with Musk’s opinion surrounding the topic, the tech expert did make some valid points that leave us to ponder."
1/3 view, as implied by conditional probability.
no 1/2 because the only information she got is i woke up at least once witche is a necessary result whether they had one face or the other witche means she has no information she should respond 1/2
@@derpderpina1804 3:18 "Before she was put to sleep, but AFTER she was told everything about the experiment..."
Nope, we are asking her at a point in time when she is completely aware of the experimental protocol.
Lemme ask you this: Imagine the experiment has been run 1000 times. What is the mean number of times the will be woken up?
Answer: 1500; 500 following H tosses, 1000 following T tosses.
Question 2: She is asked what she thinks the result was, every awakening exactly once and recordings of all of those askings are stored. Pulling out a random recording out of that set of 1500, which case is more probable?
Answer 2: 2/3 that the random awakening is after a T toss, as opposed the 1/3 after heads.
As the maker of the video said, she has enough information to transition to this objective observer statistical view. Such transition is the whole point of rationality. She is correcting for the bias incurred by the H toss being comparatively hard to observe.
(Edit: Forgot to add timestamp.)
Monday is a given. It has nothing to do with the coin flip. The same thing happens either way, so it's not chance. Tuesday is 50/50 chance.
Tuesday always happens though. This is the problem with the split analysis. To sleeping beauty she is not aware of how many times she is asked or what day it is when she's asked. It's impossible for her to draw any clues about the coin that was flipped, so without any additional knowledge, she should simply just say the likelihood of coin that was fair flipping heads is 50-50, because that is the only information about the flip she has access to.
Introducing a logical error to make the problem a paradox is nonsense. The probability is always 1/2. But if you count 1 for each Heads and 2 for each Tails, and play 1 000 000 times, the probability that the "number" of Tails be higher than the number of Heads is very high. Counting two for Tails is what is done with Sleeping Beauty by waking her twice if it's Tails, giving her twice more chances to be right (2/3 instead of 1/3) by choosing Tails.
In other words, if the question is what are the probabilities that Sleeping Beauty be right when choosing Heads or Tails as the outcome of the flipped coin, the answer is obviously 1/3 - 2/3 since if she is right choosing Heads, she is right once but if she is right choosing Tails, she is right twice. But that is a different question. The real question is what are the probabilities that the coin went up Heads or Tails, and the answer is and will always be 1/2. The "paradox" is based upon the confusion between the probabilities of an outcome and the probabilities to be right when predicting that outcome.
If I flip a coin right on my desk and ask you what is the probability that it is now (after flipping) Heads or Tails, you should answer that the probability is either 1 or 0 for each. That is very different from the probability you be right guessing whether it's Heads or Tails, because this is an event yet to come.
Something must be wrong with this puzzle because this SHOULD be a math problem. It not being a math problem tells me that some bad design is at work.
There is a 1/2 chance that the coin came up heads, being woken up is guaranteed to happen regardless
The paradox always lies in how the question is asked. Heads and Tails are merely two categories of possibility. The heads group and the tails group. Once you're IN the tails group it really doesn't matter how many times you wake her up because that DECISION point is now over. The question should be what do you think are the odds that you were or will be woken up twice? Those odds are 50 / 50. Or what are the odds that you are in the 1st half of the tails experiment, 2nd half of the tails experiment or only outcome of the heads experiment??? Then the answer would be 1/3. This isn't philosophy or math it's just a riddle with hidden information and turns of phrases. I mean... it's cute, but it's not as interesting as even the game show three doors question is. That one actually deals with a little bit of mathematics and doesn't rest on twisting words to create a paradox that doesn't exist.
Que?
Wll I think she actually can't determine the probability becouse she is put in a situation where she doesn't know the conditions to determend the probability.