If you want to vote by liking/disliking the video: “Agree with me” means 1/3 and “Disagree” means 1/2. Latest update (Nov 23, 2023): 217,332 agree with me, and 97,502 disagree with me.

"What coin? What are you talking about? Where am I? Who are you?"

As a Canadian, I would be quite happy for a 20% chance of winning against Brazil

When I reached your poll, I didn't understand the controversy. If the question is "What is the probability that the coin WILL be heads?" the answer is 1/2. If the question is "What is the probability that the coin WAS heads?" is 1/3. These are two completely different questions. The first has to do with flipping a coin. The second is about what day it is.

My reactions when I see a Veritasium video. Amazed by the title-> Understands the concept-> Trying to understand deeply-> Gets lost-> Forgets what was the video about-> Perplexed about the reality-->Video ends->Hits the like button.

I'm a simple man. The probability of everything is always 50-50. It either happens, or it doesn't.

Whenever there's no consensus in probability puzzles like this one, it usually does boil down to subtle disagreements about what is actually being asked, not the answers themselves.

What would I say? I would say “the question ambiguous, please clarify”

I'm changing my answer to 1/2 - it doesn't matter how many times she is woken up the probability the coin came up heads in the initial toss remains the same. The experiment where you count your coin toss and then mark the outcome as either Monday heads, Monday tails, Tuesday tails - is seeing the chance of being woken up by a heads flip. This is different to the probability of the initial flip.

If sleeping beauty was asked "What's the probability the coin came up heads?", I think she should say 1/2. If she was asked "What's the probability that you've been woken up as part of the outcome of a heads result?", I think she should say 1/3. I think the key thing with this question and the reason there isn't (and probably can't be) consensus comes down to how it's communicated and how we as individuals interpret what's being asked of us with the answer. If your goal is to reinforce your understanding about how the coin works, you are probably a halfer. If your goal is to be correct in answering the question from the perspective of sleeping beauty, you are probably a thirder.

Lessons learned: never let a researcher put you to sleep and never pay them in cash

I am a halfer. Using the original scenario with 2 vs 1 you could say the odds of Heads are 1/3 but that would be incorrect. There are two probabilities at work here, not just 1.Te coin flip AND the particular instance of waking up. For any particular waking moment it could be the 1/3 Either She is being woken up on 1) the only heads up 2) the 1st Tails 3) the 2nd tails. IF she was asked which one of these it is then, yes, 1/3 would be the odds but she was not asked that However, She was asked "was the coin flip heads or tails?" The odds of that, whether you have 2 wakes for tails or a million is still going to be 1/2

The correct position in this "paradox" is: (A) All conjectures about "the odds of" or "the probability of" are meaningless UNLESS they are coupled to the notion of "expected outcome over a long number of trials" in some kind of chain of events. Therefore, absent that coupling to reality (or even imagined reality), no question about these conjectures has meaning, and no answer to such meaningless questions is "Correct", and no answer is "Wrong". (B) The correct answer to the question "Expressing your answer in terms of the expected outcome of a long series of coin-flips, what are the 'odds' that it lands 'Heads'?" is "50%" i.e. "1 out of 2" or "half". (These are not different possible correct answers. Each of them is the same UNIQUE correct answer, expressed in different words.) (C) There is one and only one Correct answer to a third question. The question is longer. It is: QUOTE: Assume that Sleeping Beauty never makes a mistake in reasoning or calculation. Assume that Sleeping Beauty's definition of "the odds of something happening" is the rationally anticipated outcome of re-running the same circumstances for many trials, where 'outcome' means 'the proportion of the trials in which the event in question happened vs. the proportion of the trials in which it didn't happen". Assume that Sleeping Beauty has a net incentive to make a Correct guess. (1) Then, under the assumptions above, if Sleeping Beauty is woken up and mentally formulates a guess (whether she is asked to speak it aloud or not) as to the result of the coin-flip, and she formulates that guess based on correct reasoning and calculation, what will her guess be? (2) What will be her rational estimation of the odds of that guess being "Wrong" (bearing in mind that she has a particular way of giving meaning to the expression "the odds of", even if she knows there will only be one run of this experiment)? And (3) if she answers truthfully about her own thoughts, what will she say if asked what she computes to be the probability that guessing that the coin landed "Heads" will turn out be a "Correct" guess? UNQUOTE The third question's ONLY correct answers are: (1) "Tails"; (2) "1/3"; and (3) "1/3". To make this into a Paradox we have to avoid a lot of specifics in the question above, such as nailing down exactly what is meant by the phrase "the odds of something happening". We also have to obfuscate the fact that "What are the odds that the coin landed 'Tails'" and "What are the odds that you're going to be correct if you guess 'Tails' when asked 'how did the coin land'?" are NOT, ABSOLUTELY POSITIVELY NOT, the same question! Once someone has gotten rid of a lot of the specifics, it enables them to pretend that the question described in Part (B) above is the same question as described in Part (C)(3) above. Since each of those two questions has an obvious Correct answer, it can then be alleged that the same question has TWO correct (but mutually-exclusive) answers. That's how these char'la'tans COOK UP these phony "paradoxes". Nobody is really in the "HALFER" camp or the "THIRDER" camp. Those who say "I'm a 'halfer' are simply mis-casting their position on Question (B) above as their position on this alleged "paradox". Those who say "I'm a 'thirder" are simply mis-casting their position on Question (C)(3) above as their position on "the paradox" (which doesn't in fact exist). If you ask me Question (B) I say "1/2". If you ask me the Question (C)(3) then I say "1/3". Since those ARE NOT the same questions, I'm not vac'ill'ating or contradicting myself. My two answers are mutually consistent.

The experimenters look on in horror as the coin rests upon its edge. They somberly pull the sheet over Sleeping Beauty's face. After an appropriate period of silence, Erwin asks, "You guys wanna put my cat in a box with an unstable nucleus, a hammer, and a vial of nerve gas?" "Not again, Erwin..."

“Do not hit the like button” 87 people instantly ignored him

I think the question asked is really: "is this a heads day or a tails day?" and the chance of it being a tails day is twice as high as it being a heads day.

If I were sleeping beauty, the question of what the probability is that a fair coin came up heads is easy to answer, 1/2. That cannot change (excuse the pun). But if I'm being asked what is the probability that I'm in one of three possible scenarios - where the coin turns up heads, the answer is 1/3. Therefore, both answers are correct.

"Waking up on Monday with head" gets me every time.

There's a hidden lesson here about imbalanced classes in a dataset. Halfers are trying to model the distribution of the data generating function, while thirders are trying to minimize some loss function for the estimator.

In my opinion the question you seem to ask is: What is the probability that today isn’t Monday? The probability of a coin flip is always a coin flip as we say(50-50).. but the question refers to the coins state only in regard to what day it might be.. meaning that if it was H she’ll be asked once, while if it was T she’ll be asked twice, so it’s not really a philosophical problem, just that the question appears to be misleading

My take: The information she is told at the beginning of the experiment is all she knows whenever she's awaken. She doesn't know if she's been awaken before. The only information she gets is "Oh, I'm awake now" but not "Oh this is the first/second time that I'm awake" when she's awaken, which is something she already knew, that she'll be awaken. So going by Bayes' Hypothesis, there is absolutely no new information being added to the problem when the question "How strongly do you believe that it was heads" is asked. In other words, the state of the system has not changed from before sleeping to after sleeping because there is no new information added. Plus, the coin toss was conducted independent from any factors, and it was conducted before anything happened too (ie before she was put to sleep). So the question really is, "What is the probability it tossed Heads given she is awake?", which is no different from "What is the probability that it tossed Heads?" because her being awaken and the tossing of the coin are independent. She'll be awaken no matter what, heads or tails. (What would be new information would be whether it's a Tuesday or a Monday, which is never imparted) Going mathematically: P(heads | I'm awake) = P(heads) = 1/2, no matter what.

As a Canadian, I'm really thankful you gave Canada one in five odds of winning against Brazil 😂

The fact that she knows the coin is flipped just once means the answer must be 50%, irregardless of when she is awaken. Then again, if we are in The Matrix, then she must realize that there is no coin, and it was she who was flipped.

So based on the wording of the problem, I would say the following: When she wakes up on Monday and is asked "What do you think the probability is that the coin landed on heads?" then it is 50% because whether the coin landed on heads or tails, she was going to be woken up on Monday. Now if she is asked on Tuesday or any other day that isn't Monday, then the probability that the coin landed on heads is now 0% because the conditions she was given was that if the coin lands on heads she'll be woken up on Monday and if Tails then the she is woken up on Monday and Tuesday (and whatever day the iteration of this problem decides). The coin isn't flipped for each concurrent day. It is only flipped on Sunday to determine if she is woken up on just Monday, or if she'll be woken up on the days following Monday.

Veritasium uploaded: 0 People Agree With Me, 0 Disagree

The secret to this problem is that it is a trick question attempting to ask 2 different questions at the same time. Attaching probability to it just makes people think there is something more profound happening.

Ultimately It's a problem of linguistics and not philosophy. The question "how likely is it that the coin turned up heads" is different from the question "how likely is that you were woken up because the coin turned up heads" because the consequences of two possible outcomes of the coin are different.

It seems to me that what is proven is that Objective Reality and (each) Subjective Reality are 2 different things that both objectively exist (100%). (1/2 -> objective reality 1/3 -> subjective reality)

I think the question is subtly mixing up the probability distribution of the coin toss with the probability distribution that the sleeping beauty was woken up with a certain coin toss. So it really comes down to what you think the question is asking for.

My guy just asked a sleeping beauty problem and just left me on a thought about multiverses. I love this channel.

The question "What do you believe is the probability that the coin came up head?" has a single answer: is 1/2. The right question should be "What do you believe is the probability of SHE GUESSING that the coin came up head?" has another single answer: is 1/3. Those are two different events. The first event is "tossing the coin" with a distribution of 1/2 for heads and 1/2 for tails.The second event is "guessing the outcome of the first event" with a distribution of 1/3 for heads and 2/3 for tails.

If they ask me on Monday, I will say it was 1/2 because it could be heads or tails. If they ask me on Tuesday I would say it was 0 because it must be tails. If they don’t tell me the day and I can’t tell what day it is, then I would say 1/2. Imagine the question we will put you to sleep and wake you up tomorrow if it’s heads or in a week if it’s tails, or every day for 7 days if it’s tails. If I have no memory of anything, the passage of time doesn’t matter. Also there is only 2 outcomes, Monday and Tuesday or just Tuesday. The answer is 1/2, and just like mentioned, if it’s heads and she’s woken up one day , for a million days, we are not asking what is the probability of her being woken up on a day, we are asking what is the probability that it was heads which is always 50.

I've gone through this, and I think I've gotten to the conclusion that I'm a halver, but only on very specific conditions. I feel like two questions are being asked at the same time and each side chooses to focus on only one of them. Halvers are focusing on, sleeping beauty is woken up, she's asked what's the chance that it had come up heads. The answer is 50%, because it:s a fair coin and regardless of the day the answer is 50%. However, thirders are answering a DIFFERENT question, which is, every time sleeping beauty is woken up, what's the probability of her being right, should she always pick up heads. She's woken up everytime, is asked which one came every time, she picks head everytime, the chance of her being right is 33.3%, but it's not because of the coin, but because they're oversampling the wrong answer. Halvers are talking about the coin. Thirders are talking about sleeping beauty.

I think its the phrasing of the question that made this controversial. What if the question were " What is the chance you've been awakened due to a head coin toss?" Then to me its obvious, its one-third. Because sleeping beauty would be awakened more times due to a tail coin toss, even if she knew it is a fair coin. But if the question were " What is the chance the coin flip is a head " (With prior knowledge that she knew it is a fair one), it then would be 50-50.

I think I got it. The initial probability is 1/2, so the probabilities per day would be this: Heads Day 1: 1/2. Tails Day 1: 1/4. Tails Day 2: 1/4. In such a case the answer would be 1/2. However, this would imply that Tails Day 1 and Tails Day 2 are separate and with a Tails flip you would be awaken one of these days, not both. But since a Tails flip does always cause you to wake up twice, the probability of Tails should be double the probability of Heads, i.e. 1/3. In other words: 1/3 because not each wake-up is distinct and equally probable, if you wake up on a Tails monday, you WILL also wake up the next day, and if you wake up on a Tails tuesday, you WILL also have woken up the previous day.

Just shows that probability is observer dependent. Externally the coin being heads is 1/2, but from the princess's perspective it is 1/3.

So I guess I think if she wants to say the actual probability, she would say 1/2, but she wants to be right more often, she would say 1/3. But does being right buy her anything? If no, I would say 1/2.

Id say it’s rather a linguistic problem: It’s a 1/3 chance that if she is awake, it was Heads. It’s a 1/2 chance that it rolled Heads when she awakens at all.

Its not that theres a 1 in 3 of heads, its a 1 in 3 that she was awoken for it being heads

For me it boils down to this - in one iteration of this test, if she wakes up on the first day when heads has landed and says tails she is wrong 100% of the time. If she wakes up on the Monday and Tuesday and says heads each time she is still wrong 100% of the time. She is not 'more wrong' simply because she has answered the same question twice (especially when from her perspective she has answered it only once). This perception that you have a possibility of being 'more wrong' and that you should hedge against it is what leads to odd psychological outcomes, for example where a room full of actors answer an obviously wrong question and the one non-actor goes along with it despite knowing, for a fact, it is wrong. Also equating this to Monte Hall is totally wrong, the reason the probabilities change is because you know, for a fact, one of the doors without the prize has been removed. Here you have no such change so I am somewhat unclear why this was drawn out as a point. Now you could argue - she knows there is a possibility she will wake up twice but that is not the same thing at all. The difference is in Monte Hall you are asked to make 2 decisions based on changing information, here that is just not the case

To me it's the phrasing of the question asked that's important. If every time she's woken up, she's asked "do you think the coin came up heads or tails", she should always answer tails, because similar to the Monty hall problem, there will be more scenarios of her waking up and the outcome is tails. But the question isn't asking her what she thinks **the outcome** is, but instead it's asking her what she thinks **the probability** is. The probability of the coin toss is completely independent of how many times she wakes up, or even if she wakes up at all, and it is always 1/2. So even if she were to wake up and the actual outcome of the toss was tails, she is still correct by saying that **the probability** of the toss is 1/2.

I think this scenario highlights, more than anything, that it’s odd to phrase a question with multiple answers with a yes or no prompt.

The issue with the simulation is that we presume that "in the not too distant future" realistic complex simulations of reality that are so convincing are possible. Yet, we have no reason as of yet to presume that such simulations are physically possible. There may be some things that seem logically possible given parameters of a game but are not physically possible in a real unviverse.

my postion is different based on question, if asked "what is the probability YOU(sleeping beauty) is awake from a coin that landed heads" Then the answer is 1/3. But if asked, "what is the chance that the coin flip that woke you up showed heads" My answer is 1/2

The dilemma is not "what is the correct answer", but "what is the question being asked?". If Sleeping Beauty is asked what is the probability the coin came up tails, her answer should be 1/2. If the question is "what was the result of the coin toss" and the challenge is to be right (significantly) more than 50% of the time, she should answer differently. In other words, the disagreement is not about what the answer should be, but about what the challenge was in the first place. The only sensible answer is therefore: Restate the question as to remove the ambiguity. Or 42. That works too. Same reason.

Teo things are for sure: 1) The probability that the coin was tails is 1/2 2) The probability that sleeping beauty has a f*cked up sleep cycle at this point is 100%

One flip of a two sided coin. 50/50. Doesn’t matter how many times she was/would be asked.

I would argue that, due to her being asked the question thrice, and two of those are in tails, the probability of it landing heads from her perspective is 1/3. But that's her perspective; the sheer probability of the coin flip is dependent on nothing, so the amount of times she is woken up does not affect the probability of the coin flipping heads, only her perspective of what the probability could be, therefore, the probability of the coin flipping heads is 1/2 but, only for her, the probability is 1/3. This is probably riddled with flaws, so please do not be afraid to call me out on them, I love to debate! Also, to give my opinion on the universe/multiverse example, the probability of there being one universe is 1/2, but the probability of you being in a universe in either the multiverse or the universe is x-1/x, where x is the number of universes in the multiverse.

I think this is more a problem with the question having multiple valid interpretations than it is an issue of the question having multiple valid answers. Halfers are focusing the question on the origin of the random event that causes a decision to be made at the start(i.e. the flipping of a coin). Thirders are focusing on the end result of the overall experiment (i.e. the number of ways sleeping beauty can be woken up). The tricky part in this whole scenario is that the question is presented as a single event with a single function to model it. However, from my perspective as a programmer, this scenario is better described as a chain or series of two functions. The first one generates a random 50-50 result (flipping the coin). That random result (heads vs tails) is that function's only output. Everyone can agree on the probability of each result for that function on its own. Now we take that outcome, and use it as the input for a separate function. This second function simply makes a decision on the number of times to wake sleeping beauty up. It becomes pretty obvious when looking at this function in isolation that its results are skewed towards the side that wakes her up more times. The second function essentially multiplies the likelihood of the input that would cause multiple wake-ups. Thus we arrive at the two interpretations of the original question and their different answers. Interpretation 1: How likely is the coin to come up heads? -> obviously 50%. Interpretation 2: How likely are you be woken up by the coin coming up heads vs tails? -> obviously 33%. Both are valid and so my personal stance on it is that the question is ill-formed by being ambiguous.

I tried to apply philosophy to probability in my Probabilities class in college and almost failed the course. So, you know what my vote is.

The question basically is "what is the chance that you woke up just one time?" So I would say 1/2

As a Math undergraduate, the thirder is answering a fundamentally different question from the original. You’re weighing each question equally, which should not be the case. To the sleeper it’s an isolated day (Monday H 50%, Monday T 25%, Tuesday T 25%). To the sleeper, the probability that the coin landed heads is 50%, but the expected value for winning $1 for each correct answer, is $2/3. To experiment on the probability rather than the expected value, you’d have to tally up only one random day per trial, rather than counting multiple days per trial. Of course the expected value would be $2/3 because there’s always $1 on the table for Heads, but $2 on the table on Tails case, but it’s a fundamentally different question from the original.

A lot of the other scenarios were not equivalent to the Sleeping Beauty scenario. They were more like asking Sleeping Beauty "Do you think it's Monday?" That's an entirely different question from "What are the chances the coin landed on heads?"

The REAL problem is that there's an implied reward: if she's asked the "probability" of heads, then it's 1/2. If she's being rewarded for GUESSING whether the coin was heads or tails, she should always answer tails, because she'll get rewarded twice in that scenario (vs once if the coin flip was heads).

This has highlighted the futility of reasoning, in both hopeful and distressing ways. On the one hand, here is a man who understands both perspectives, what purpose each perspective serves, and is able to objectively and mathematically prove what the correct answer is under each premise. And he made a video spreading that information. On the other hand, people are still debating about which perspective is categorically valid, and this guy is even encouraging that debate. All in spite knowing that its just a matter of premise and that both are provably correct.

Albert Einstein dealt with relativity based on position. (Are you watching events occur aboard a spaceship or watching from a planet?) Could probability be thought of as a problem relative to your situation? ( whether you are awake or asleep) Maybe also refer to Schrödinger's cat. The probability of the cat being alive or dead can change based on your situation for the event. Being inside or outside of the box.

My thought process for picking 1/2 is as follows: The coin is flipped only once. In the Tails scenario, both wakeups originate from a single coin toss. Since the coin is fair, the question if heads was up would be 50:50 for me. In my mind, there's no "third option" like shown on the paper (4:18), because whether its monday(tails) or tuesday(tails), it's still the same coin toss. If we sort by heads/tails instead of monday/tuesday, we have heads(monday) or tails(monday/tuesday). Now, if we rephrase the question as "What's the probability you were woken up because the coin landed on heads", then it's 1/3, because only 1 out of the three total wakeups originates from heads.

This problem is more of a word problem than a math problem. As i worked through it my understanding of the problem grew and as such my answer changed. The question "what is the probability the coin came up heads?" is two questions, depending on how you parse it. I think thirders and halfers are both correct and wrong, because they're answering different questions. One side is answering the probability of the coin turning up heads/tails when it was flipped. The other side is answering the probability of you being in a state where the coin came up heads vs tails. They're different problems with different solutions. What is the probability the coin came up heads? 50/50. What is the probability i will be right if i guess heads? 1/3rd.

The probability of an observer's "choice" being "true" changes based on the knowledge of the observer. Pretend there are three observer's in the snow white problem. 1. Snow white. Before the experiment. 2. Snow white. During the experiment on an arbitrary morning. 3.The coin flipper. Knows the results of the coin toss but always keeps the results a secret. 1. The coin flipper starts in a secret sound proof room where they flip the fair coin. After flipping the coin, the coin flipper asks their self a question, "Based on my knowledge of the situation, what do I believe the probability that the coin came up heads?" The coin flipper says to himself EITHER, "1/1, I just flipped the coin silly!" OR "0, I just flipped the coin silly!" 2. After the coin is flipped, the coin flipper comes out of the secret room and asks snow white, "Based on your knowledge of the situation, what do you believe the probability that the coin came up heads?" Snow white replies, "1/2" 3. Later than evening snow white is put to sleep totally forgetting about the conversation due to the spell. Every time snow white is woken up she is asked the same question by the coin flipper: "Based on your knowledge of the situation, what do you believe the probability that the coin came up heads?" And each time she replies, "1/3" This is somewhat of a semantic issue that may depend on how we define the word probability and how the word probability is being used. For example we could interpret the word probability to refer the fundumental truth and reality of the situation, however we could also interpret the word probability to be based on the observers knowledge of the situation. These are two totally different things.

One of the best philosophical thought experiments I've heard in a while. Thanks for sharing.

The key difference is that the event where she is asked about the coin happens twice with the same answer if it comes up as tails. The question is worded in a way that no matter what her perspective is, the answer is 1/2. However, if the question was instead: "If you guessed heads, how likely would you be to be right?" Then the answer would be 1/3 because she gets asked the question once if she's right, and twice if she's wrong.

Your ad was perfectly timed after you had us “go to sleep” 😂 As soon as the screen went black, it cut to Patrick Stewart’s smiling face telling the camera, “Hello, I’m Patrick Stewart”

I would propose to modify the football/soccer match as follows, since this, at least for me, makes it a bit clearer: If Brazil wins (80% winning chances) you are woken once on a Monday. If Canada wins, you are woken 30 times on a Thursday. This experiment is repeated every week. Now, when asked, which day it is, what would you answer? If the goal is to answer as many questions correctly, you should answer „Thursday“. If you want to be right as many weeks as possible about your day choice, you should say „Monday“. So, in the end it boils down to the frame of reference not being clear in the original question, and without that clarification, all answers are dependent on your choice of what that frame of reference is. At least I hope so…

Before watching, she should answer 33%. of the 3 times she wakes up, only 1 of them was due to heads being the result. So in 2/3 of the situations in which she is awake, the result was tails. All 3 options of her being awake have equal probability of occurring (They all have a 50% chance of occurring, but 2 occur together). So on any given result when she is awake, 1/3rd of them will be a result of a heads flip, and 2/3rds the result of a tails flip. Thus she should answer that there is only a 1/3rd chance that she is awake due to it landing on heads.

Veritasium: it's a fair coin Sleeping beauty: is it fairer than me ? Veritasium: yes, we are living in a simulation Edit: wow thanks for the likes . Actually I was confused between the snow White and the sleeping beauty. Snow White is the fairest of them all. That's why she got killed

When he said "Don't hit the like / dislike button" , exactly at the same time my like and dislike icons in UKposts started "Glowing" .....what is that ? Magic?

The difference between the coin and the soccer game is that the coin is fair, both sides have a 50/50 chance. In the soccer game example it's not 50/50, Brazil is a lot more likely to win than Canada.

This whole problem seems like someone asking, "If something happened, but you dont know that it happened, what are the chances it happened or not?"

The problem with doing the vote this way instead of a poll is that so many people are going to ignore the beginning and like the video because they like the video and not because they agree.

5:26 - Reaching into a bag of one white marble and million black marbles is a fundamentally different exercise: the black marbles could be selected independently each other, but if Sleeping Beauty happens to have been woken up on day number 1000000, then she must also have been woken up on day number 999999, and the day before that, etc. Days two to one million are conditional on day one.

I have the ambitious position. What we are talking about is actually two questions. Both 1/2 and 1/3 are correct, but they are answers of different questions

The simulation example also has multiple philosophical perspectives. We currently don't have examples of a simulation indistinguishable from reality, so we are either the first simulation in a series of simulations yet to create a simulation, or we are the last simulation in a series of simulations yet to create a simulation, anything in between and we would have simulations that replicate reality perfectly, so the "Are we in a simulation?" question collapses back to a 66% at best as the possibilities are 1. We aren't in a simulation. 2. we are the first civilization in a simulation. 3. we are the latest civilization in a series of simulations. Of course the argument can be made to collapse this again and say we are either in a simulation where we haven't created a simulation, or we aren't in a simulation and it doesn't matter if we are first or last in that series, giving the reality/simulation options each a 50/50 probability. Both these results have a similar nature to the sleeping beauty paradox depending on perspective. Finally there is an argument that if a civilization can create a simulation to replicate it's own civilization, then that simulation will fairly quickly grow to create it's own simulation, and so on, so there are an endless series of simulations, the likelihood that we are at the start or end is so small, that is almost impossible for us to be in a simulation and not have our own simulation when compared with the idea that we just aren't in a simulation.

I think the real question is: does it matter what Sleeping Beauty thinks or how many times she’s right? The probability of the coin toss is still 50/50. EDIT: It seems (like most logic questions) that this is really a semantics issue. Is it: probability coin is heads based on it being flipped once, or based on which way the coin is facing up when she wakes up. So we’re not really learning any deeper truth to the world with this question, it’s just a matter of was our specific setup properly explained

The confusion arises from the same term “probability” being used for two different things: 1. the probability of getting heads when a coin is flipped (50%); 2. the probability of Sleeping Beauty in her confined situation guessing correctly if she believed that the coin had come out heads (in the past!). SB’s chances are, of course, skewed to tails. On Tuesday she may only guess when it had been tails. Had it been heads she would sleep and could not guess. In other words her guess entails her own dependence on the coin. Imagine you are lying on the operation table: The doctor tells you that you have a 50% chance of dying and never waking up from the narcosis. But what should you assume after you wake up? That the doctor comes and tells you: “Sorry, I goofed-you’re dead!” ??

5 stars Derek. I think your whole channel is excellent. But a few of your videos stand out as gem mint in terms of provoking deeper thought and exploration. Thank you!

The coin only gets tossed once, it is always 50/50. The relevant conditions. We have a fair coin. It is flipped once. The other conditions are not relevant to the question. If you ask on Monday it is 50/50 probability, if you ask on Tuesday it is 50/50. We ask a person about the probability of a fair coin landing on heads for a one time event. It does not matter that you asked the same question multiple times, this is independent of the coin flip. This is not the Monty Hall question. What day it is does not have any relevance to changing the probability. It will always be 50/50 because you are not asking any other question such as 'what is the probability you think the coin landed on heads but it is really Tuesday?' 'What day is it?' You are not asking her to pick the correct answer. You are asking for the probability of one flip of a fair coin only.

Derek: Okay so the game is about to start and you fall asleep... *ad starts to play and shows you a product that claims to help you sleep better* Me: Simulation theory sounds just about right.

I think the _really_ clever thing here is that Derek has carefully orchestrated a video to generate a high "like" _and_ "dislike" count. That kind of controversy will be irresistible to the almighty algorithm 😎

The trick to this paradox is that when asked about the probability of the coin being heads, the answer depends on whether you believe implicitly baked into that question is also what day of the week is it. If you were to flip a coin on Monday and Tuesday, the answer would be 1/2, but that’s not what’s happening. Instead, the coin is only flipped once and then there are three possible scenarios because you get woken up twice under tails. The other factor has to do with that probability is based on an assumption that you’re repeating an experiment infinite times basically. In other words, when we say that a coin flip is 50/50, that’s assuming you flip a coin infinite times. You could flip it 1000 times in a row and get tails every time theoretically, and if you base your conclusion off that you would say that coins are 100/0. We know that isn’t true though. Similarly, if the sleeping beauty experiment was repeated infinite times, the results would show that in 1/3 of cases of her being awoken, that the coin landed on heads.

i dont understand the halfer's perspective The question is " what do you belive is the probability that the coin CAME up heads? " ( now that u are awake) Heads - u get one awakening Tails - u get two awakenings If u have already been awaken and asked this question, 2 times out of 3 it was because it was tails, so the chances that tails came up and as a result u were awaken and asked this question, is 2/3 (or 1/3 heads)

The question is the trap as you explained in the video: "What do you believe is the probability that the coin came up heads?" You would have to disagree and ask them to clarify if they mean: "What are the chances of a fair coin flipping heads or tails?" OR "What are the chances you are in either stage of waking up in the experiment". Edit: i hate these kinds of "math problems" since they are almost always about the question being asked in a stupid/inaccurate/unfair way to the situation at hand and then people just going "what if we actually try to answer the unfair question seriously". Then it inevitably ends up with the same conlusion as the first paragraph where the authors assume one of X interpretations of the question and continue to calculate and answer that. But in that case you could have just asked the correct question from the start in the problem. This is why I always tell my friends to think about what they are saying, if it can be understood in mulitple ways it wont help you get your point across. Write so that your intention can only be interpreted in one clear way.

I am on the 1/2 side, here is my reasoning: The many wakings are not independent events (I think). If any one of them happens, you are certain that all of them will happen as well. Because of this, I think you can imagine all those sequential awakings as a single event. That removes the bias of "many possibilities in one branch" and you are left with single event in each branch. edit: This also makes sense to me for the argument of "no added information", because the sequential wakings are linked together.

I say it's 1/2 because...first, the probability of being woken on Monday P(monday) is 1. Whether heads or tails she'll still be woken on Monday. Second, the probability of waking up on Tuesday P(tuesday) is 1/2. Therefore P(heads)= 1- P(tails)= 1- (P(monday and tuesday))= 1- (1*1/2)=1/2

Waking up is a condition signal for Heads on Monday, Tails on Monday and Tails on Tuesday , we have a total of 6 possible wake up signals thus , 1 signal for HH, 2 signal for HT , 1 signal for TH and 2 signal for TT , P(Signal = H ) = 2/6 ,P(Signal =T)= 4/6 .

Here's my way of thinking about it: We make 100 people go through the experiment and for each person we flip a coin and act accordingly. Obviously, we expect 50 of the coins to come up Heads and the other 50 to come up Tails. When we ask a person about the probability that THEIR coin came up Heads, it is equivalent to asking about the probability that they belong to the group of people whose coin came up Heads, which is 50%. Thus, the answer they should give is "50%". This process is equivalent to making a single person go through the experiment 100 consecutive weeks, on half of which the coin would come up Heads and on the other half it would copme up Tails. Still, it is the same conditions, the same question and therefore it should be the same answer of "50%". However, let's now consider what would happen if every time we wake up the person we ask them "Do you think the coin came up Heads or Tails?" and they ALWAYS answer "Heads". If we do this for 100 weeks, then on 50 weeks the coin will come up Heads and on the other 50 the coin will come up Tails, as expected. But for every one of those 50 weeks when the coin came up Tails, we ask the person twice. In total, we ask the person 150 times and only on 50 of those the coin came up Heads. Thus, they answer correctly only 33% of the time. In the first case, which is the one presented in the video, the answer of "50%" is always correct INDEPENDENT of whether the coin came up Heads or Tails, since that's how fair coins work, they have a 50-50 chance of coming up Heads or Tails. In the second case, the individual is correct 33% of the time, given that they always answer "Heads" and that they get asked twice every time the coin comes up Tails, but that was not the question at hand. Overall, I believe that the answer of 1/3 is answering a different question to the one that is actually asked in the experiment, as presented in the video, and that's why it seems correct to some, however the best answer that someone part-taking in the experiment should give is 1/2.

Okay, I actually gave it some thought before listening to the proposed answers, and initially I also answered 1/3, but then I read the comments, did some more thinking, and switched to 1/2. First, as others said, the wake-ups are not independent, and the question comes down to "which branch do you think you're in", and the probability of that is 50%. Second, even in the situation with 1 vs a million awakenings, you can't consider being in each of them equally likely. Finding yourself in the branch with a million awakenings is dependent on the actual coin toss (50%), and the probability of finding yourself in a specific position in the second branch must be calculated using Bayes' theorem. 50% it is.

Let's base this question of the marble example. Given no knowledge about the prior coinflip: What is the chance that you receive a white marble. That would indeed be 1 in million. Now you are told a coin was flipped for either receiving a white marble or a million black marbles. The chance of either happening is still 50/50. It does not matter how many marbles or times you are put to sleep. As this has no influence on the coin that was flipped beforehand. Source: I follow the highest educational system until adulthood for probabilistic events in mathmatics. When veritasium noted down the amount of times he either woke up on monday or monday, tuesday. He should've noted the amount of times he trew either heads or tails, which would forecast how many times you are woken up. However how many times you are woken up does not influence the past, which has already happend.

To me, similarly to how in your counting experiment you doubled up the tallies for Tails every time, it's 50% because the tails numbers are being arbitrarily inflated by double-counting. It's similar in my mind to if you said "toss a coin. If it lands on Heads count it as one, but if it lands on Tails, count it as if it happened twice". The coin is still 50% we're just counting it wrong (in my opinion, which isn't worth much).

For me, it is the wording of the question that tells me 1/2. "What is the probability" is a different question than "which outcome do you think happened".

What is actually being asked? The question asked of her is, "What do you believe is the probability that the coin came up heads?" How should she answer? 1/2. The question asked of her is about the result of a single coin that was only flipped one time, not about her waking up or going to sleep or anything else. She doesn't have any information to help her figure out either the result of the coin flip, or the odds changing in any direction, so the odds are the same as if she were predicting in advance if it would come up heads. While It's true that it is all about how you interpret the question, the question right there for us to examine and question the correct way to interpret it. Enough information is present in the question to know the correct and incorrect way to interpret it.

I think the problem with counting results on the paper, is you are only marking the success. If you write Succes or Fail for each trial, you get twice as many chances of getting it right on TAILS, but you are also getting twice as many chances of getting it wrong….

i love how just when i feel certain that im right he makes me question it

Waking up on Tuesday always comes after the coin flips Tails. There is no other reason for this event. If we want to correctly relate the question to the situation it's incorrect to put more variables into the equation such as "there is one more day in which she may be awake". The question has to be only related to the thrown of the coin which possibility is 1/2.

The issue here, just like the monty hall problem, is how the question is asked. If you think about it this way, it should be obvious: if the coin is flipped tails, it is guaranteed that she will be awoken on both Monday and Tuesday. So, if A is the situation where we get heads, B is when we get tails on Monday, and C is we got tails on Monday, then woke up Tuesday we can see that P(A)=P(B), but also P(B) must also equal P(C) because if you get tails you are guaranteed to be put to sleep and wake up again tuesday. So, P(A)=P(B)=P(C).

As a halfer i find this problem to be the biggest problem with humanity. We are more concerned of being right, than getting the answer right. Seek the truth, not validation.

A fun problem where the two answers are actually answering two different questions! The skill is not figuring out which is right, but understanding how the two questions are subtly different. Good thinking exercise and excellent video as usual.

For my part, I think fundamentally the question is malformed, and that's why we have such issues with it. There are two possible meanings of the question, and commensurately two possible ways of looking at the data. A: "What do you believe is the probability of the coin landing as heads?" B: "What do you believe is the probability, given you are awake now, that the coin actually was heads on Sunday?" The ways of looking at the data (if we treat it as sampling whether Sleeping Beauty thinks the coin actually did land heads/tails): 1. In each _trial_ of the SBP, which answer will be most consistently correct? 2. For each _awakening_ of Sleeping Beauty herself, which answer will be most consistently correct? If we base our statistics around the per-awakening result, then 1/3 is correct, and indeed it should be 1/(n+1), where n is the number of times you awaken Sleeping Beauty if you flip tails. If we base our statistics around the per-trial result, then 50% is correct. The former is true because, when we look at the percent likelihood *on any given awakening* that Sleeping Beauty was awakened on a trial that flipped Heads, that of course must fall to zero as the number of Tails-awakenings tends to infinity--the vast majority of awakenings are Tails-awakenings. That the latter is true is a bit more complicated, but can be expressed as follows. Perform the same test, but simply ask SB whether she actually DOES believe the coin flipped heads, yes or no. Tally up the answers. If the coin _did_ flip heads, then she will either be right once or wrong once. If the coin _did not_ flip heads, then she will either be right N times (for the N awakenings), or she will be wrong N times. All told, there are 2N+2 possibilities, and out of them, (N+1)/(2N+2) = 50% are correct. Hence, it depends on whether you examine the data from a per-awakening basis or a per-trial basis. The question is malformed, ambiguous, and that is why it leads to an alleged "paradox."

The question still deals with a single coin toss. So it has to be 50/50. The issue of her waking up multiple times on a tails outcome is a macro event from the coin toss. Correctly guessing the coin toss from her perspective is still 50/50 each morning. She's just got more opportunity to play this twisted game on a tails result.

The answer is one-half. It doesn't matter how many times she's woken up. If the question were, "What day is it?" well then she should say Monday, because she's guaranteed to wake up on Monday. I think he pointed out the key issue well. The question changes the probability because you are talking about different event spaces. The event space for a coin flip is divided evenly between heads and tails, namely the probability is a half for both. The event space for "what day is it?" is split 0.75 for Monday and 0.25 for Tuesday. By the total probability theorem, the initial coin flip partitions the space equally. So 0.5 for heads and 0.5 for tails. Assuming the coin flipped tails, there is equal probability that it is Monday or Tuesday. P(Monday) = P(Monday | Heads) * P(Heads) + P(Monday | Tails) * P(Tails) = (1 * 0.5) + (0.5 * 0.5) = 0.75 P(Tuesday) = P(Tuesday | Heads) * P(Heads) + P(Tuesday | Tails) * P(Tails) = (0 * 0.5) + (0.5 * 0.5) = 0.25 The value of one third as a probability does not factor in anywhere here. The odds for it being Monday is a third, but not the probability. Most importantly because the coin flip does not depend on the day. Changing the number of times she wakes up will obviously change the probability, but Monday will always have a slightly higher probability. Now the situation changes if she is given a score for guessing heads or tails. Let's say she gets a point for getting the answer right. In this situation, the number of time she wakes up will effect which answer is better. In the case of getting woken up Monday and Tuesday, the answer doesn't matter, the most she can score is one point. But if she's woken up Mon, Tues, and Wednesday when the coin flips tails, then she could score up to 3 points. This changes the event space we're talking about. If Sleeping Beauty wants to score more points, then she should say Tails because it will pay out higher in the end if she's right. Interestingly, the expected value of this random variable depends on her strategy. If she only says Tails, she will average about 1.5 points in repeated iterations of this game. If she says Heads or Tails equally, then her payout will be 1 point on average. I think this falls into Game Theory territory, so not sure if someone with more experience can elaborate on that.

This is what happened to me during my school exams. I would forget what I studied the previous day. The teacher would think the probability I studied for the test was 1/10 from the marks.