Veritasium - 2023-02-11
For decades, the Sleeping Beauty Problem has divided people between two answers. Head to https://brilliant.org/veritasium to start your free 30-day trial, and the first 200 of you will get 20% off an annual premium subscription. ▀▀▀ Many thanks to Dr. Mike Titelbaum and Dr. Adam Elga for their insights into the problem. ▀▀▀ References: Elga, A. (2000). Self-locating belief and the Sleeping Beauty problem. Analysis, 60(2), 143-147. - https://ve42.co/Elga2000 Lewis, D. (2001). Sleeping beauty: reply to Elga. Analysis, 61(3), 171-176. - https://ve42.co/Lewis2001 Winkler, P. (2017). The sleeping beauty controversy. The American Mathematical Monthly, 124(7), 579-587. - https://ve42.co/Winkler2017 Titelbaum, M. G. (2013). Ten reasons to care about the Sleeping Beauty problem. Philosophy Compass, 8(11), 1003-1017. - https://ve42.co/Titelbaum2013 Mutalik, P. (2016). Solution: ‘Sleeping Beauty’s Dilemma’, Quanta Magazine - https://ve42.co/MutalikQ2016 Rec.Puzzles - Some “Sleeping Beauty” Postings - https://ve42.co/SBRecPuzzles The Sleeping Beauty Paradox, Statistics SE - https://ve42.co/SBPSSE The Sleeping Beauty Problem, Reddit - https://ve42.co/SBPReddit Sleeping Beauty paradox explained, GameFAQs - https://ve42.co/SBPGameFAQ The Sleeping Beauty Problem, Physics Forums - https://ve42.co/SBPPhysicsForums ▀▀▀ Special thanks to our Patreon supporters: Tj Steyn, Meg Noah, Bernard McGee, KeyWestr, Elliot Miller, Brian Busbee, Jerome Barakos, M.D., Amadeo Bee, TTST, Balkrishna Heroor, Chris LaClair, John H. Austin, Jr., Eric Sexton, john kiehl, Anton Ragin, Benedikt Heinen, Diffbot, Gnare, Dave Kircher, Burt Humburg, Blake Byers, Evgeny Skvortsov, Meekay, Bill Linder, Paul Peijzel, Josh Hibschman, Mac Malkawi, Mike Schneider, jim buckmaster, Juan Benet, Ubiquity Ventures, Richard Sundvall, Lee Redden, Stephen Wilcox, Marinus Kuivenhoven, Michael Krugman, Cy 'kkm' K'Nelson, Sam Lutfi. ▀▀▀ Written by Emily Zhang, Derek Muller, Tamar Lichter Blanks Edited by Fabio Albertelli Animation by Ivy Tello, Fabio Albertelli, Jakub Misiek Additional video/photos supplied by Getty Images & Pond5 Music from Epidemic Sound Thumbnail by Ignat Berbeci Produced by Derek Muller, Petr Lebedev, Emily Zhang
As a Canadian, I would be quite happy for a 20% chance of winning against Brazil
fax
Amen
In the particular universe that produces Canadian dominance of Brazil in soccer...pigs can fly. Pig guano everywhere.
As a Brazilian, I’m happy there’s at least one scenario where we are more likely to win against Canada
@@diegototti we are also more likely to win against canada in a war. they would apologize for being involved in a fight and raise the white flag
"What coin? What are you talking about? Where am I? Who are you?"
I thought something similar at first too, but actually it is all carefully crafted to prevent this from being a valid answer. It is only when she is put "back to sleep" that she forgets, and what she forgets, is being woken up. So every time she is asked the question, she remembers the original explanation, the original time being put to sleep, and being woken up that time.
That would logically mean X, but I don't like X, so it doesn't mean X.
Great science right there, chief 👍
Wait, why am I naked?
@@stilldreamy5181This comment still makes sense, since the paradox itself stands because it introduces "knowledge about the system", which leads to the "simulation theory" aspect of the question. If you are exclusively a part of the system, meaning you can't imagine a system containing the system you are in, then the question will have only one valid answer; it's when you assume the possibility of a "hyper-system", a system that contains the system you are in, that the question becomes a paradox.
Therefore, questioning the reality of a "system within a system", like the original comment does, is the key to "solve" the paradox. In other words, the hundreds of papers discussing this paradox, are really people debating their belief in the multiverse or simulation theory (which is actually unprovable, therefore a theological debate).
Professor Farnworth response lol
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.
Agree.
I like the way you explained this. His statement “Something changed” was important because it matters that an event occurred between observations.
This is the correct answer. It depends on how the question is interpreted.
But if sleeping beauty doesn't remember any times she's been woken up, every time is her first. So to her it's always 50/50. Any other wake-up (Tuesday) in her existence never actually happened
I think there should be a distinction between asking "What is the probability A coin came up heads?" and "What is the probability THE coin came up heads?" The question is about THE coin, and given she is awake, the answer is the probability of her being awake.
Derik: vote with like and dislike button
Youtube: makes dislikes invisible
Fr 🙄
71% of viewers liked the video. I just checked :)
@@poseidonsmafia1160 Wow how did u know that??
@@nezukoyaegerr
There are Chrome Addons that let you see dislikes. I use "Return Youtube Dislike".
@@poseidonsmafia1160 it doesn't let you see actual dislikes though, it only shows those by people who also have that same extension
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.
Yeah, it just seems like semantics. I depends on whose perspective you are using
Semantics, asking the wrong question, wrong definition, etc.
That's what makes Monty Hall problem so great - it's not about words, it's about the actual concept itself.
Yeah, "what was the probability that it came up heads?" vs "what is the probability that it came up heads?" can already make a difference to the answer. Only if you define questions properly can you answer them. I suppose that's why they were philosophy papers and not mathematics. In mathematics you need things to be defined unambiguously.
There is clearly a majority consensus on the entire thing with most people leaning towards the real world side instead of the fairytale book side. Why do you think they use a literal fairytale character to point this out? Math is 100% disconnected from reality. A concept. She's literally missing 25% of her ability to know what actually happened. She is at 75% comprehension of her reality since she can't tell the difference between waking up once or waking up twice. But the knowledge shown to her is letting her know, that she has two chances to respond on a tails flip, or once chance to respond on a heads flip. So she can take the chance of being right or wrong about a 50/50 chance twice in a row, or once. Her best chance of answering correctly on monday heads, monday tails, or tuesday tails is to realize that there is no tuesday heads and eliminate 25% of her ability to answer. Thus leaving 3 equal chance scenarios. Her real world probability is skewed by lack of information. Her fairytale probability is 1/2, because 1/2 is 1/2 and everyone knows 1/2 is 1/2.
I'm a simple man. The probability of everything is always 50-50. It either happens, or it doesn't.
exactly
Reminds me of the football coach who didn't want his quarterback to throw because two of three possible outcomes were bad. Interception and incomplete.😅
@Glenn Clark ahh the coach is wrong. It's still 50-50. The pass either reaches the teammate or it doesn't. 🤣
Average mulla thought process
No. It’s not. If that was true, you would win any game every second round on average making only random choices, i.e., tossing a coin. Clearly, that’s absurd.
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.
Thank you
Exactly. Seems fairly obvious
But what if it was a 1% chance ( a 1 on a 100-sided die) to wake up 1 million times? Even if you were asked what the probability of the die being 1 WAS, it was still only 1%. Its unlikely that you were put to sleep a million times in the first place.
@@alexs1277 Not clear what variation you are describing here. But If 99% to wake once, and 1% to wake 10^6 times, chance die WAS 1/100 is 99.990%. No?
Changing the tense of the question has no impact. Again just consult the soccer game analogy. It’s obvious
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.
Same
That’s just his fans
real
"Good Morning, Beauty. What do you think was the probabilty for the coin to be heads?"
"50%. Can I keep the coin?"
"Sure."
"What time is it?"
"Well..."
opens Harry Potter's Gringotts vault
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..."
Split the difference!
@@ChrisContin They divvied up the hadrons amongst themselves and Erwin got a new cat.
Ahh I dont have enough neurons in my brain to understand this, someone please do the honors.
@@bluzter it is a reference to Shrodinger’s Cat
Wait, hidden result, hidden protocol. Then they wake her up -1/12 times and Poland wins the Super Bowl
“Do not hit the like button” 87 people instantly ignored him
Now 2,479...
How are you able to see number of dislikes?
@@Varma17 the title is the amount of likes (agree) to dislike (disagree)
@@Varma17 I think the title updates periodically, the number of people disagree
@@Varma17 there are browser plug-ins to show dislikes again. the one I use is "return youtube dislikes". cheers
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.
Then take them both to the consideration and calculate the average. That would be the real solution to this dilemma.
@@orka6848 no, these are not two approaches to the same question, they are two different questions. Averaging them is kind of meaningless.
Estimating the distribution is not the same as minimizing expected error.
@@johnmorrell3187 I think you hit the nail on the head - those who agree with him are answering a different question than those who do not.
funny, but no: the imbalance of the heads and tails here is only due to a deliberate mistake in sampling; because of a sampling error you record "tails" twice when a single "tails" event occurs, but only a single "heads" event is recorded for "heads" events. The dataset is seriously screwed up; when presented with a new "instance", the "thirder's classifier" will have its probability estimates wrong: it will be predicting "tails" with prob. 0.66 but it will only be "tails" with prob. 0.5.
@@orka6848 here we have the engeneer
For me it becomes less paradoxical when I think of the question as rephrased as "How likely is it that Heads is responsible for you waking up this particular time?"
that is so much better of a question, kudos
Exactly dude, like it's simply not possible for it to exist. 1/2 is simply the answer.
"what's the probability it came up heads" contains a hidden knowledge assumption, that being with the knowledge that it just woke you up or prior to that being someone's state. It's a partly a matter of how you understand the meaning of the question, but the best answer is based on all knowledge, which is that you just woke up. Imagine she's never woken up for heads. She's woken up, what was the probability that it was heads? In an abstract sense 50:50, but based on the full knowledge of evidence, it's zero
Yes, indeed, the answer to the question in the video is 1/2 and the answer to your question is 1/3 (the 'intended' answer)
Your question is simply different from the original
Lessons learned: never let a researcher put you to sleep and never pay them in cash
That would logically mean X, but I don't like X, so it doesn't mean X.
Great science right there, chief 👍
the researcher just wanted to kiss sleeping beauty multiple times
@@AuGAlaNstud
"Waking up on Monday with head" gets me every time.
Best way to start a Monday
That's why I pick heads everytime
Some people prefer waking up with tail.
By Veritasium? I'd only want it to be from Sleeping Beauty. If not, I'd pass
bruh
As a Canadian, I'm really thankful you gave Canada one in five odds of winning against Brazil 😂
As a Brazilian I'm thankful for 4 out of 5... Canadian team is getting better and better (Brazilian team have been a lot better).
And there's a 100% chance of another balloon flying over Canada will be shot down by an F22. :D :D
As a Croatian, we beat you both, even though Brazil was better but unlucky against us. It was that 1/5 win for us 🙂
Good luck to Brazil!
@@lukatore123 - I think it was more like 2/5. Croatia's got a great team (maybe the best one per capita - amongst Uruguay and Portugal). Brazilian team, of course, had better individual quality, but Croatia had a very interesting collective game.
Afterall, i think it was very well deserved
@@BillAnt 🤫🤫🤫
Its not that theres a 1 in 3 of heads, its a 1 in 3 that she was awoken for it being heads
In half of all trials over 1 billion simulations, she will wake up because it is heads on day one. 50% of the pie is spoken for . If you agree that the odds of it being heads or tails on day 1 are 100%, then there is only 50% of that probability remaining for all results starting with tails on day 1. There cannot be a 2/3rd’s probabilities stemming from that 50%.
It's 1/2. ;)
@@hellomate639 Don't really want to argue so i will just throw my opinion on board, and there are 2 posibilities for coin landing, either heads or tails, but if tail lands she will be woken up and askes question 2 times and if heads lands, it's once, so she wakes up 3 times and asked the same question same question with only one being correct and as such there is 33% chance that answer is heads and 67%. Tho I thought it's 50% at start too, but tails have 2 outcomes and because of that odds are not even.
@@striuk4259 It's so painful having a reasonable explanation as to why it's 1/2 but needing to save it.
I actually have experience with probability theory, so understand the 1/3 position, but I do think it's wrong for very concise reasons.
@@hellomate639can you please explain it
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.
Yeah I agree, it's more about semantic than statistic. Derek just found a nice trick to get tons of likes and views with a question that is more intellectual masturbation than anything else.
@@En_theo exactly. And I love Derek and his content but this video just felt like a gotchya. And the worst part is I can't even express this to him by downvoting the video
Maybe it's a social experiment on how much influence his opinion has
Exactly
I agree it's a trick question, but it's not two different questions. It's just one invalid question. The tail scenarios cannot be viewed as two separate outcomes: informationally they are identical to sleeping beauty, and therefore the same outcome. The question just arbitrarily labeled the tail scenarios as two outcomes, not with any kind logic compatible with reality, but with memory erasing magic.
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.
The formulation of the question directly tells you to consider it from sleeping beauty's perspective.
In other words, if we repeat the experiment every week for the rest of eternity, is she trying to be right most on days or right on most weeks ?
I really like how you worded this. And you're 100% percent correct. I personally believe that because of the way that the question was asked that it should be answered from sleeping beauty's perspective just as @rantingrodent416 stated, but the way you acknowledged both points of view without hating on either one I very much respect.
Flip heads, put one green bean in the bowl. Flip tails, put two red beans in the bowl. You pick a bean, what are the odds it is green?
@@rantingrodent416 Well she has no way of telling if she was awaken or not, so her only guiding point would be her understanding of the fact that a coin has only two outcomes, so it would be 50%. If someone flips a coin and ask you what are the pobability of it being heads, with no previous context (as sleeping beauty didnt remember if she had been awaken) you would answer 50%, because there is no way for you to say how many times you have been asked that question.
My guy just asked a sleeping beauty problem and just left me on a thought about multiverses. I love this channel.
The probability that she guesses the side of the coin is ~1/6. ~1/2*1/3=1/6
But if you ask about the probability objectively, then of course ~0.707
It has no corelation to multiverse unless it exists (probability of Multiverse unknown)
@@aucklandnewzealand2023 stop dude you're talking to an anime pfp
@@aucklandnewzealand2023 Honestly, that isn't just as justifying seeing that she could have done any other operation
@@roddraft3466 the general consensus is that your pfp doesn't affect your comment
The whole thing can be solved by just acknowledging that there's two possible answers because of the unclear wording of the question rather than arguing over which is right while knowing that the question is ambiguous
Exactly. It’s just a matter of clarity. If you allow for more scenarios in your question, then there are more possibilities.
I love how this comment has four likes but is sandwiched in-between comments with 1k and 1.4k.
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.
Not it’s still 1/3 when she awaken because she awakens twice if it’s tails
It's a fairly complex situation, but I agree completely. If you jump to a conclusion you are ignoring the actual dilemma, which is how semmantics may affect our perceptions of the universe. There's no truly correct answer, only a correct answer given a chosen context.
You wanna know the probability of heads vs tails? 1/2
You wanna know the probability of Sleeping Beauty correctly guessing if today is Tuesday? 1/3
etc
Makes me think how much of actual science is affected by linguistic biases, I would guess most of it.
It’s always the language that is the issue in these kind of paradoxes. Write this problem using only math and suddenly there is no paradoxes
I disagree. It’s a 50 50 chance if when she’s awake it’s heads or not. It’s a 50 25 25 chance if she is waken when MH, MT, TT respectively, because it’s 50/50 whether it’s head or tails and then if tails 50/50 whether it’s Monday or Tuesday.
Wittgenstein is proven right yet again
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.
Yeah, one of the confusions is that "what's the probability that the coin came up heads" can mean different things. Halfers think it's a question about the behaviour of coins. Thirders think it's a question about your on-the-spot beliefs about past events.
@@AzrgExplorers I agree. Thirders actually think that the question is, "what are the chances that you were woken up once before?"
yup, like nearly all things, the readers interpretation is what truly matters... and yet the world doesnt care
@@wordsofcheresie936 No, thirders are answering the exact question asked. Sleeping Beauty wasn't asked "did the coin come up heads?" She was asked, "what are the chances that the coin came up heads?" In the soccer analogy @veritasium used, he talked about this difference without actually pointing it out.
About ten billion humans have been born. So the odds of you being born as you is one in 10 billion. So when I ask if you are you, what is your response? If I ask what were the chances that you would be born exactly as you are, what is your answer?
The questions are different and so the answers are as well.
The best way to explain it is the way he already did. Let's Make a Deal gives you 3 doors, with only one valid prize, heads. The other two have tails behind them. Then they take away a confirmed wrong door, giving your probability of choosing heads an increase. That's why you always switch the door you choose after the removal of a tails door.
This method is simply presenting you with two possible doors but then adding a 3rd confirmed possible door. Your safest bet is to be realistic and realize that the original two doors always had a 1/3 chance of having heads no matter what door you chose. Changing doors still results in a 1/3 chance of choosing the heads door.
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%
Underrated comment lol
I like how you state that the chance is a half as one of the two things that are 'sure', despite the dozens of scientific papers with discourse, this video, the other comments, and the whole nature of this debate. Guess you had the answer all along then.
2/3
You are incorrect about #1.
The probability that the coin was tails is either 0% or 100%, depending on its result.
@@feha92 thats actually true no joke, since he specified "was tails" and anything that happened in the past either happened or didn't happen
"Do not hit the like button..."
The like button: GLOWS AND TEMPTS YOU
Veritasium uploaded: 0 People Agree With Me, 0 Disagree
0.3 is probability of one side of the coin vs 0.707 probability of the other side
Well no sh*t
@@aucklandnewzealand2023huh? Where does that come from?
@@aucklandnewzealand2023more like 0.33 and 0.66
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.
"what is the question being asked?" is not a dilemma. The question is clearly about "the probability that the coin came up Heads". Answer to that question is 50%. And I agree with you that those who answer 1/3 are answering the wrong question.
that is so perfect an answer. how did you make it so easy,, in that, what is your background?
@@jonathanlavoie3115 what is the challenge being set, then. Is it to answer correctly on what the coin toss was, or something else?
That's the dilemma here - not what is the correct answer, but what is being asked of her in the first place.
If the challenge was « guess the outcome and I give you 1$ » she would answer Tails, not because the probability is 2/3 but because the reward is twice. Just like I give you 1$ if you guess Heads right, and 2$ if you guess Tails right. You would answer Tails not because the probability is higher. It remains 50%. In the SB experiment, the question is the probability it came un Heads.
@@uRealReels Thank you. You're the first person who reply to me so kindly!
A short anecdote about me:
In my programming course there was an exam in probability and statistics. Three of the questions were about the same problem. In a basket containing 9 blue balls and 11 red balls, what are the probabilities of A) draw 2 blue balls. B) 2 red balls. C) 2 balls not the same color.
Questions A and B are very easy. But for question C I knew that the teacher wanted us to use a complicated formula learned by heart. I didn't want to use this formula because 1- The formula is complicated and I'm lazy, 2- I don't like to use a ready-made formula that I don't fully understand and 3- I wasn't sure if the formula really applied to the situation.
So, I solved question C by following this simple reasoning: Probability of 2 blue balls + probability of 2 red + probabilities of 2 different = 100%. Total must be 100% because there is no other possibility. As expected, the teacher's formula answer was not the same as my answer, and I had to argue to get the point, but he had no choice but to acknowledge that his formula didn't apply to the situation, and that my answer was correct.
I argued my point in front of the review board, not because I needed the point (my average was already 98%) but because I like the truth. That's who I am...
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.
I've reasoned about this and I think it is correct to say 1/2. In my opinion 1/3 is simply wrong because it is not equally likely to be in any of the three cases. I'll copy here what I already said in other comments that are lost in the haystack.
My opinion: When she is asked about the probability, the coin has already been flipped and its state is determined even if unknown to her. So here the word "probability" should be interpreted as her confidence that the coin landed heads. She is aware of the procedure and she knows that the coin is flipped one time at the beginning. Imagine she is asked the question immediately after the toss (of which she doesn't see the result) before being put to sleep. She would obviously answer 1/2. From now on there is no reason she should change her initial guess because the coin is tossed once for all and there is no subsequent event that could influence the output. It doesn't matter if it's the first or the millionth time she's being awakened: because she doesn't know what day it is she never gains new information and there's no reason she should update her initial guess.
1/3 is simply wrong because it assumes that the probability of being in one of the three cases is uniform while it is not. The probability is actually 1/2 of being Monday and it landed heads, 1/4 that is Monday and it landed tails and 1/4 that it is Tuesday and landed tails.
The 1/3 argument moves from the wrong assumption that to the question "what day do you think it is today?" she should be 2/3 sure it is Monday. Actually she is instead 3/4 sure it is Monday to balance for the fact that there is no Tuesday/Heads combo. The probability it is Tuesday is in fact P(it landed tails) times P(it is Tuesday | it landed tails).
I put video at 0.25x and he made a terrible error in his experiment. Look for yourself what he does. He simply writes a sign two times when the coin lands tails. He should have tossed the coin a second time to decide where to put ONE sign. If you do it right you get the expected 50-25-25 proportions.
I wanna add something to make it more intuitive: in the case she is awakened 1 million times if it lands tails the probability that in any awakening that day is the first Monday is about 50% and not about 0%. Think of it this way: if she is asked "what day do you think it is today?" she is better off answering "The first Monday" because is much more likely to guess it landed heads and hence surely it is the first Monday than to guess it landed tails and then identify one of the million possible days.
Shocking take
The probability of the coin flip doesn't change with the way we want to measure it. If Sleeping Beauty was woken up a million times for a tails flip, it wouldn't make the coin flip any less likely to turn up heads. Being woken up two times instead of one doesn't make one outcome twice as likely as the other, as the thirder perspective implies. If we're asking about the probability of the coin flip alone, like the question in the video (1:04) very clearly is, then the answer cannot be anything other than 1/2.
Now, if the question was anything like "For N times Sleeping Beauty was woken up, what is the probability of her being woken up because of a heads flip?", then it'd clearly be 1/3.
Let’s do a little thought experiment: I tell you: „I‘m about to flip a coin. If, and only if, the coin flips heads, I‘ll call you.“ The next day, I call you and say: „I flipped the coin now. What do you believe is the probability that the coin came up heads?“
What would be your answer?
I know it sounds counterintuitive,but the only correct answer for sleeping beauty is 1/3.
When she wakes up, there are three possibilities: A: heads/monday, B: tails/monday, C: tails/tuesday.
Obviously, A and B have the same probability, because it’s a fair coin flip, so if they would repeat the experiment every week, she would wake up every monday and the coin would have flipped each side 50% of the weeks.
The probabilities of B and C must also be the same, because every week she wakes up on tails/monday, she also wakes up on tails/tuesday.
So the probabilities of all three possible outcomes are the same.
And the sum of the three possibilities must be 100%, because A, B and C are the only possible outcomes, and each time she wakes up, only one of them can be true.
Thus, the probability of A: heads/monday is 1/3.
P(A)+P(B)+P(C)=1 and P(A)=P(B)=P(C)=1/3
The more I think about this, the more it becomes a question "what is conciousness?" and "do you believe in reincarnation?" rather than just a probability question.
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.
Maybe that was the real point the originator was trying to make but people just totally missed it and now here we are
My first reaction was that the problem is too contrived to be interesting.
actually, the probability of it being monady or tuesday is 33 percent, but the odds of it being tails is 50 percent
@@Sad_cat_studio no the odds of it being Monday is 66%
That would logically mean X, but I don't like X, so it doesn't mean X.
Great science right there, chief 👍
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.
Facts I don’t get how the root problem is that complex or controversial lol
@@kuribohoverlord2432 cause you are a genius mate, congratulations
What if the rules dictated that she would only be awakened and asked the question if the coin flip game up tails? Then, there would still be a 50-50 chance that the coin flip was heads. But given the information that she was being asked the question, she would know that the coin flip was not heads.
The fact that she is being asked the question gives her additional information.
What makes this "controversial" is that some people are unwilling to adjust their beliefs when given new information.
It's still just a matter of what's meant by the question. If a flip a coin, and you see that it's heads, and I ask you, what are that chances the coin landed heads, there are two answers depending on how you interpret my question. Either you answer 50% if you take my question as "what was the chance of what you've just seen occuring in general" or 100% if you interpret my question as "what is the chance that what you saw (the coin landed heads) is the actual state of the world (the coin landed heads)"
Thank you. You phrased it beautifully
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
Veritasum: now, will you eat the red apple or the blue apple
Prince Charming: [kisses Sleeping beauty]
Researcher 1: Stop! you're wrecking the experiment!
Researcher 2: Interesting, this proves we live in a disney simuatoin.
@@blankregistration7301 Or do we? 🤔 * suspenseful veritasium music *
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.
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.
That's hilarious. I dominated that class because of multiple degrees in philosophy. And went on to teach deductive, inductive, and probabilistic logic. And intro to inductive and probability logic class is pretty much proving the laws of statistics and much harder than any statistics class I ever took. Stats prof definitely hated me tho.
Philosophy begins where science ends
Or is it the other way round
@@nyjsackexchange
Mathematicians and physicists were philosophers at one time.
@@aglawe1 Science is born of philosophy
The scientific method begins with a question
@@nyjsackexchange
So it is an iterative process, philosophers ask questions and scientists try to answer them.
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.
Knowing Derek, The like/dislike options is a study in of it's self. We'll get another video where the like is the wroner answer and then a later video examining the results.
@@brandonfrancey5592 That makes a lot of sense. I'd bet that is the actual purpose of this video.
I liked this question as a vote to the proposition that people expressing enjoying the video will have a massive distortive effect on any attempt at polling.
(Edit: Wait don't use comments as polls! Dislikes just bury the poll itself!)
I have liked your comment because I agree with it.
I am pretty sure he knows enough scientific methodology to know this liking/disliking thing is complete bs.
It helps increase interaction so I guess it's a smart trick
Bob watches Alice flip a coin.
The coin comes up heads.
Alice: "What do you believe is the probability that the coin came up heads?"
Bob: "What do you mean? That the coin has come up heads? That the coin would've come up heads? That the coin comes up heads in general?"
Alice: "Pick one."
This isn't a math problem. It's linguistics.
Considering how important it is to write formulae correctly in math, it does feel like not just a linguistics problem but one that's intentionally vague so as to make two different outcomes seem equally obvious.
Good point. Once the coin has already landed, it's no longer 50/50, it's either 100% or 0% heads.
Specifically, it's semantics
Yes and no. This is just a rephrasing of the Monty Hall problem.
@@kirbwarriork3371 Yep it seems like a lot of information is left out just to create a clash between two different answers to two different questions.
“Scientists have calculated that the chances of something so patently absurd actually existing are millions to one.
But magicians have calculated that million-to-one chances crop up nine times out of ten.” - Terry Pratchett, Mort
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 is exactly it. I feel like this problem wasn’t really posed thoroughly and that causes confusion
@@hisuianarcanine9379 Nah, the question was clearly "What is the probability that the coin came up heads", that fits perfectly the first case of OP and it's unambiguously 1/2
Exactly what I think! It's unclear which question is being asked from this video, and we have to be very specific when asking the question.
Like you said, if the question is "what is the probability that the coin landed on heads", the answer is always 0.5. Sure, sleeping beauty will be wrong multiple times if the coin landed on tails, but that's not relevant to the question being asked here. The fact that the coin is two-sided does NOT change, and the sleeping beauty's knowledge is identical every single time.
It's an entirely different question if sleeping beauty is trying to 'win' as many times as she can, then the best answer is quite obviously tails. If she is woken up N times when tails is thrown, and once when heads is thrown, she will get the correct answer N times out of N+1 guesses, on average when repeating the problem.
@@angivaretv4475 "The probability that a fair coin comes up heads" is undoubtedly 1/2, but "the probability that you are waking up on a heads monday" is 1/3.
@@myeloon It would be a waste of everyone's time to go into the exposition of her being waken up monday monday/tuesday if all we are going to ask is that given a random fair coin that is absolutely irrelevant to her situation. Of course the question is "given that you were just woken up, what is the probability that heads came up". If that is not the intention, I hope the people who developed this problem and six degrees of separation from themnever wakes up again.
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
Right. The important part is "she doesn't remember any times she's been woken up." So every Tuesday her may we well have never happened. To her it's always the first wake up, which to her is 50/50.
But that wasn't the question. Thats the entire point (in my opinion) of this thought experiment: There are additional parameters at play (how often she is woken given a certain outcome) and given those parameters what are the odds? Put it another way: What if heads doesn't wake up? Then whenever she waked it will be 100% tails, even tho the coin has a 50/50 propability.
She is either correct or incorrect she is answering with a probability.
right? doesn't change that she'll wake up twice, it's not as if the coin is being flipped again everytime she wakes up. it's just that if one happens one set of events happen and if another happens a different set of events happen. no matter how frequent .
Exactly, the extra steps to validate a non function was just mental gymnastics, but after listening to it once more a coin flip is a coin flip aka ½
When he said "Don't hit the like / dislike button" , exactly at the same time my like and dislike icons in YouTube started "Glowing" .....what is that ? Magic?
AI when it hear like it glows
Dude you can't repurpose likes/dislikes for a poll. That's now how things work.
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.
EXACTLY, probability? heads, obviously, what you think the result for this run was? tails, obviously
My thoughts exactly! Was looking for this argument.
What is the probability of coin came heads - 1/2, because that is the fact.
What is the probability that we woke you because coin came heads - 1/3 and is very different question.
What I was about to type.
but she wasn't asked what is the probability a toss of a coin comes out heads. She was asked what is the probability the coin did come out heads.
There is a big difference in asking about the probability of an event that has not occured vs the probability that a specific event has happened in the past so long as you gain knowledge when transition from that past point to the present.
One view the point when asked what is the probability of A. Which is 50%
What is the probability of A|B (A given B in statistics).
The probability of A given I have information B modifies the probability of A having occurred.
This is not an independent probability but a dependent one.
i agree with this because fundamentally she can't remember if she been woke up before (according to the experiment) so the fact that she is awake now can't be used to bias the answer dose 50/50 should be the right answer. correlation does not equal causation.
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.
agree with this, but would say I'm a halfer in this instance because the exact question asked is 'what do you believe the probability of the coin being heads?' not 'what do you believe the probability of being woken up by the coin being heads?' subtle difference, but to one question I'm a halfer, the other a thirder.
Danm
@@superkeefo6951 This. That question sounds to me like question that would be asked in a hospital to check if my brain functions correctly like what's the date, who is current president etc. It made me 1/2er just because of semantics but I understood what he meant and in that context I'm 1/3er, so I don't know whether I should like or dislike
@@0NeeN0 but if you're saying there is context then you are essentially adding it and rephrasing the question given to you to be the second question. That's the point momo was making, the implied context makes you think you need to answer the second question. But really the question should be asked with that context or else it's 50/50
This! 100% this! The problem is that the language being used isn't precise enough.
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.
What if we change the problem, such that if the coin lands heads, she is never woken up. If the coin lands tails, she is woken and asked the question. In this situation, it's the same as if she can still see the coin on the table showing tails. The probability is 100% that the coin landed tails.
Bruh that's the same question
If she's asked EVERY time she woke up, then it'd be 1/3 because two times when asked, it had been tails. If she was asked only once, decided by the coin flipper, then it should be 1/2.
I don't think it matters to rephrase the question. If she had a record of how many times she guessed the face of the coin correctly through trial and error she would get to the probability being 1/3rd for heads. But this is only because she doesn't know if its Monday or Tuesday. So I agree with the first part of what you said. She you and I know the coin toss 50-50. But what's asked if is it's actually heads when she wakes up. This actually flips the assumptions around where it becomes obvious it should be 1/3. But I think people misunderstand what 1/2 would actually mean. It means that because she has no connected information between the time she wakes up the probability remains 1/2. So to believe 1/3 means you believe her inability to have information about waking up a second time is information.
@@Furiends wording matters. P(head) and P(head|awake) are two different question, the video seems to be asking the former
The blog "Less Wrong" has the definitive breakdown and solution of this. Briefly, it turns out that the most common arguments in favor of either position are wrong and are subtly solving a different problem. The Thirder argument (due to Elga) proceeds as if one of the possible awakenings (Monday&Heads, Monday&Tails, Tuesday&Tails) is generated at random and you are asked for the probability of heads. But in the original problem, there is no justification for randomly sampling the possible awakenings as independent, equally-likely events. On the other hand, the Halfer model due to Lewis treats the problem as if tails led to one awakening, either Monday or Tuesday, which produces incorrect betting odds.
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.
true.
dislike guys. we have to rise.
That's the whole point..... The events are conditional from researchers perspective.... From sleeping Beauty's it's not....she doesn't know what number of time she has been woken
But any time you go to sleep, you either wake up and are asked about the coin flip, or wake up and are told the experiment is over. The fact that you're being asked about the coin flip is information, and it tells you that the experiment hasn't ended. Knowing that the experiment takes longer to end if you flipped tails, you use the information to update your probabilities.
Well said!
@@elliotgengler3185 You could also take this scenario in a darker direction by slightly altering the experiment, whereby a 'heads' coin flip results in Beauty never waking up at all. If she is then awoken to be asked about the coin, she knows that the coin must have been tails.
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.
Agreed. It is the perspective.
"Came" is the keyword. It's past tense. When an event has already occurred, any information you can access regarding that event changes the probability that it occurred one way or another. What's the probability that the card I pulled out of the deck is the ace of spades? 1/52. But now you draw a card. It's not the ace of spades. Since you've removed that card from the list of possible cards I might have, the probability that the card I pulled at the beginning was the ace of spades is now 1/51.
Since it's a past event, new information about it changes the probability.
I draw another card. Now it's a 2/51 probability that I have the ace of spades. You draw another. One less possible card I could have, so now it's a 2/50 probability that I have the ace. And so on and so on until all the cards have been drawn and the probability becomes 1/1 whether I have the ace or not.
I do think at least for those fully understanding it that it's about how we value information. Thirders are incorporating the fact she lacks information. Halfers are assuming lacking information is irrelevant. For the sake of Halfers it's important we define the problem of her guess in one single instance based on the rules. There is inaruguably three states in the state space. She's awake on a Monday with tails, she's awake on Tuesday with tails and she's awake on Monday with heads. I actually think it's Halfers that have one extra step of justification. (unless you completely missed this is about the shared information that she lacks information.. it's not a matter of perspective). That extra step is say even though she knows there's three states in the state space there's ultimately only two that matter. The third being she doesn't know it's not Tuesday and heads so the question is like saying it's 50-50 on Monday or Tuesday.
It's pretty clear that he asked the first question, it's explicitly written on the screen. So thirders are just wrong.
I agree. And as a halfer I have to point that the question is "What is the probability that the coin was heads"
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.
Targeted ads. It means stop wasting your life on youtube and go to sleep. Rofl.
Had an ad for a Canadian University for the football match lol
Same , I mean not the same ad but perfectly timed
I think I got a good example for this:
Let's say 50% of new born child will be a single child and 50% that it will be twins. So there is 50% that you an single child right? But after 100 births there will be (on average) 50 people that are single child's and 100 people that are twins, so that's 150 total people, so if you ask a random person if he is a single child, there is actually only 33.3% that the answer will be yes.
No. There's always a one third chance that you're a single child, because even after only one birthing, it's either single or twins. You presumably don't know if you are a twin or a single,,so you have to answer one third. Unless the question is "what are the chances of this birth being twins?"
@GeeEee75 ya that's what I said
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".
Exactly. This is an independent event. Probability conditional on being awake though, I think that's different although I'm not sophisticated enough in probability to know how 😂
@@aubreydeangelo Not necessarily. The claim of them being independent is contentious among theories of probability. According to Bayesian probability theory, probabilities aren’t objective; instead, they reflect our degree of belief in X given Y information, so the totality of our information on the scenario actively affects the “probability” in the epistemological sense of an outcome. The existence of objective probabilities is tenuous at best; Quantum mechanics wave function collapse is a possible exception, be it contested. They are of course competing frequantist theories of probability however it being independent is not at all intuitive or obviously true.
Exactly, the question is ambiguous. There are 2 questions being conflated.
- what is the probability that a fair coin came up as heads this week?
- what is the probability that we woke you up because the coin came up heads?
@@nicksmith9521 thank you! None of the research papers would be necessary if the question was specified.
@@nikhilweerakoon1793 There's no need to tap into some subjective probability nonsense. There are two probabilities at play.
Implicitly, the question is stringing together two dependent probabilities: (1) a 50% chance of turning up as heads, and (2) 100% more likely to wake up due to Tails
Let's use another example: I flip a coin. 50% chance it's heads and I wake you up. 50% chance it's tails and you die in your sleep. The next day you wake up. "What is the probability it landed on heads?"
The probability is 100%. Because you have been woken up. The coin flip was a 50/50 chance, but the waking up was a 100/0 chance.
Yes, flipping the coin in general is a 50/50 shot at heads. But now that you have more information (the fact that you woke up), you need to factor that in. If you say "50% chance" because the independent coin flip had a 50% chance, you're just intentionally ignoring additional information in some kind of weird linguistic purism.
@veritasium - 2023-02-11
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.
@unnamed5338 - 2023-02-11
ok
@aftabahmed-_- - 2023-02-11
👍
@AnasKhan-fq8yb - 2023-02-11
First
@-nary-zy5jt - 2023-02-11
I disagree with u.
@neopiscator2095 - 2023-02-11
😅