The Most Controversial Problem in Philosophy
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- Опубліковано 10 лют 2023
- For decades, the Sleeping Beauty Problem has divided people between two answers. Head to brilliant.org/veritasium to start your free 30-day trial, and the first 200 of you will get 20% off an annual premium subscription.
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Many thanks to Dr. Mike Titelbaum and Dr. Adam Elga for their insights into the problem.
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References:
Elga, A. (2000). Self-locating belief and the Sleeping Beauty problem. Analysis, 60(2), 143-147. - ve42.co/Elga2000
Lewis, D. (2001). Sleeping beauty: reply to Elga. Analysis, 61(3), 171-176. - ve42.co/Lewis2001
Winkler, P. (2017). The sleeping beauty controversy. The American Mathematical Monthly, 124(7), 579-587. - ve42.co/Winkler2017
Titelbaum, M. G. (2013). Ten reasons to care about the Sleeping Beauty problem. Philosophy Compass, 8(11), 1003-1017. - ve42.co/Titelbaum2013
Mutalik, P. (2016). Solution: ‘Sleeping Beauty’s Dilemma’, Quanta Magazine - ve42.co/MutalikQ2016
Rec.Puzzles - Some “Sleeping Beauty” Postings - ve42.co/SBRecPuzzles
The Sleeping Beauty Paradox, Statistics SE - ve42.co/SBPSSE
The Sleeping Beauty Problem, Reddit - ve42.co/SBPReddit
Sleeping Beauty paradox explained, GameFAQs - ve42.co/SBPGameFAQ
The Sleeping Beauty Problem, Physics Forums - ve42.co/SBPPhysicsForums
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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
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.
ok
👍
First
??th
I disagree with u.
"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?
you are either a mechanical engineer or a philosopher 👍
@@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).
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
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.
aandd you became a thirder
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
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.
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
What would I say? I would say “the question ambiguous, please clarify”
No the question is clear, after being woken up what are the odds that the coin flip landed on heads/tails. The answer's 50/50.
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.
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.
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.
I changed due to this
I agree.
Exactly my thoughts! "Tuesday tails" simply doesn't exist, if we're talking about coin flipping.
I'm actually changing my answer to 1/3 because while a coinflip is obviously 1/2, the way the problem is constructed is : heads gets only one question, while tails gets 2 questions (questions not flips!) , so by saying tails you are going to be right twice as much just because you get 2 questions instead of 1 . IF the coin would have been flipped again between monday and tuesday, then the answer would obviously be 1/2. This way it's just 1/2 heads and 1/2 tails, but tails gets double points every time.
Change it again :) Imagine that if heads, she's not waken up at all. She's only waken when tails. She's asked the same question. What should she answer 1/2 or 0?
It's 1/2, think of it like this. You randomly get assigned a red or blue magnet, which you cannot see. They only attract coins of the same color.
You are given a bag with 1 million blue coins and 1 single red coin. You will then stick in and pull out a coin with your unknown color magnet.
What is probability that reaching in with the unknown color magnet that it will pull out a red coin?
If the random 50% chance assigned you a red one you will therefore will pull out the 1 red coin.
If the 50% chance assigned you a blue magnet, therefore you will pull out a blue coin, which will be one of those 1 million blue coins. The 1 red coin will always be pulled every time yours is red, but any specific blue coin (outcome) will only be pulled one / one millionth of the time yours is blue, the 50% blue probability is split amongst the one million blue coins you have a random chance at pulling, so those specific outcomes has far less weighted probability than the red coin. The red magnet's 50% will never split its probability because there's only 1 coin. More blue coins added just further dilutes the probablity of any specific coin being pulled from the blue magnets probability, but the sum will always still just be 50%.
“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
"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
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.
That's not correct. No matter which day it is, she's got a 50% chance of being right. We don't compound the tails days together. It's just as likely that she's waking up on the Monday from heads than it is she's waking up on either Monday or Tuesday from tails.
@@TheShepdawg9 The difference is that if tails came up, they are asked twice. That changes the likelihood of heads vs tails when asked
@@TheShepdawg9 she can litterally not be asked anything if it's both Tuesday and heads. This means that the probability of the coin having fell on heads on a Saturday is 0. So yes, the day it is when the question is asked does influence the probability of the result.
No, it’s “is this a heads sequence of days (1 day long) or is this a tails sequence of days (2 days long)”.
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.
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
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
@@bluzter The cat referenced above, plus they're saying that they flipped the coin and it landed on neither heads nor tails, landing instead on its edge and therefore she will never awaken. It's the hidden third result.
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
I'm extremely happy you posted this content!
Pure creative thought in action is a beautiful art form! ...and more importantly there's so many perspectives on this idea!
Anyway... just really happy to see someone explain this so well!
Thank you!
He did not explain it well, leading some readers pondering possibilities unreasonably based on a misinterpretation of the Sleeping Beauty scenario. If that was unintentional, then fine. However, this is not a big statistics problem.
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 🤫🤫🤫
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.
Yes, that is the single point on which this hypothesis depends. Also it's not about just "people connected to the Matrix from outside" kind of simulation. It's about the people simulated inside it, i.e. conscious NPCs like the agents, etc. Ability to simulate many conscious beings would be probably much more complicated than just simulating sensory inputs for conscious beings that already exist.
as you said at the start, it depends on what question you're asking. pretty much all probability problems are very sensitive to interpretation, and almost the only reason people get different answers is that they're answering different questions.
perhaps this also explains the people who think that 1+1*2=4 - they interpret the question of "what's 1+1*2?" differently, even though they have all the information needed to interpret the question the intended way.
edit: ok i suppose a big part of why communication works at all is that we have mostly agreed on how to interpret stuff. so the sleeping beauty problem is a problem not in math or philosophy, but in linguistics. it becomes a math problem once the linguistics is worked out, and an easy one, actually. i guess the philosophy problem is about realizing this? idk.
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
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
This is just the difference between the mean and the weighted mean. Why is it a debate? The mean of a coin flip (0 is heads and 1 is tails) is 0.5. But the weighted mean where heads is doubly weighted is 2/3. Not paradoxical at all.
In ML training, with a mse loss function, the weights for the datapoints are linearly important while the accuracy is quadratically important. That’s because the error (loss) for a single row is w*(y_pred - y_act)^2. If the objective is a random process like coin flips, the optimal prediction is just the weighted mean. But if the objective is highly predictable, then the quadratic can dominate and the predictions for weighted vs equal-weighted may start to converge.
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
So an estimate of probability is only correct if it is what the infinite repetition of a test would converge upon.
Toss a coin and you'll get a head or a tail. Toss it 10, 100, 1000, a million times, and the ratio of heads and tails will converge upon 50/50. That's how you know you got it right.
Here we're asking a princess to judge the likelihood that the coin landed heads.
If we repeat the experiment 1000 times, then we will ask the question ~1500 times. If every answer is "1/2", then they will ALL be wrong, because in reality around 2/3 of the questions will be asked after a tail.
@@peelingOn 1000 experiments, you still ask the question 1000 times, not 1500. Even if you ask 10,000 times the coin flip is still 50/50. If the coin flip was heads, it will be heads on the first ask or the 10,000 ask.
@@claytoncourtney1309 No, over the course of 1000 experiments, you will ask the question approximately 1500 times (once per iteration if it was heads, twice if it was tails).
Think about it this way:
Suppose the experiment is run 1000 times, and each waking of the princess is recorded as a separate video. So there are ~1500 videos.
One video is shown to you at random.
What are the odds that video is from a waking that occurred after heads was tossed?
It is ~1/3rd, right?
You, watching the video, have exactly the same information that she did at the time it was recorded.
So why should her answer be different to yours?
@@peelingI rethought it through, in the shower of course lol, and I agree with you. I DO like your idea of using the video.
I still have problems with the question but am already late for work. I did, once i saw your your response wanted to let you know i now agree with you.
This seems like confusion about valid states to me.
By adding the wake instances, you're artificially splitting a single outcome. TailsxTuesday is a different day of the same result, not a different result. The odds that it's Tuesday when she wakes is 1/3, but the odds of heads is 1/2.
here's one to demonstrate this:
if you get heads, i'll break one of your legs. if you get tails, i'll break both.
The likelihood of having 1 leg broken is 50% (monday). The likelihood of having two broken is 50%(tuesday). There is no 1/3. you only have 2 legs and there are two outcomes.
Breaking the second leg on a different day doesn't change this. Not knowing doesn't change the odds.
Applying odds to a wake instance is simply nonsensical. it's just a parallel state.
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 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.
Thank you for breaking my brain with that last question!
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.
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
I love coming to this channel after a pre-roll
I really enjoyed this video. Initially I was thinking that it was very obviously 1/3 but now I am a bit in between but siding towards 1/2. Thanks for the video!
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.
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
It's 1/2 with certainty. First off, outcome does not mean even probabilities. The coin flip will either be heads or tails so it's a 50% chance that it's heads and the monday, or 50% chance that it will be tails (with each of the awakened days splitting that remaining 50% probability). Run it a million times, ~0.5 million times it will be heads and a monday, and ~0.5 million times it will be tails, it's just that "what kind of" tails those 0.5 mil are is random, and not equal in probability to a "heads" outcome that does NOT further split it's 50% probability.
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.
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...
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.
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 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.
'What is the probability that you will wake up from this much heroin dosage'
If 'heads' wipe out all life on earth, and 'tails' doesn't. After the coin has been flipped and before you have seen it, someone asks you, "what is the probability of 'heads'?" What would you answer?
@@Poppillon50
@@adampicki that's the answer to the question "what was the probability of the coin toss". Not the answer to the question "what is the probability that a particular event happened". The key to this is the implied impact of the circumstances under which the question was asked.
@@Poppillon but the probability was still 50 it happened, only because it landed it doesnt make it 100%
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.
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 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
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
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
@@nyeaglesfootballgarbagemen8346
Mathematicians and physicists were philosophers at one time.
@@aglawe1 Science is born of philosophy
The scientific method begins with a question
@@nyeaglesfootballgarbagemen8346
So it is an iterative process, philosophers ask questions and scientists try to answer them.
i love how just when i feel certain that im right he makes me question it
Only a Sith is based.
@@kaizokujimbei143 no obi-wan! no!!!!
hello there
Let me make you question it again, the answer is 60% tails and 40% heads for 2 flips :)
I sense great turmoil and hesitancy in you
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 .
I stand on the ambiguous question position. I think that when you ask Sleeping Beauty about the last coin toss only, it is 50-50% of chance.
Although when you analise the whole experiment, and you ask her about it, 1/3 makes sense. This changes what you're currently looking for in the experiment and also the question asked should be more specific.
A good example for that is let's say you want to check for the probability of each day she would wake up. That's going to be 2/3 for Monday and 1/3 for Tuesday on a single check. Running the experiment more times, might change the question and the answers.
On the other extreme cases, specially when it involves weights and infinity, I think when the weights are similar they get cancelled in the "formula". But when they aren't equal, they affect the problem deeper which would answer Canada x Brazil and the infinity cases. How to build this formula of if it ever exists, I have no idea. :)
My last contribution to this is: what if, the probability is a "superposition" of values? So that 1/3 and 1/2 are a viable answer to the same question?
Probability never made sense to me, even less now. Thanks @Veritasium! :)
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.
There is no question to which the correct answer is 1/3. The whole thirder perspective is flawed because it treats the possible "states" as equally likely and independent, but they are not independent.
@@nekekaminger The question would be, ‘What is the probability you were woken up by a flip of heads?’
I think the answer to the sleeping beauty question is 1/2 though because like the original comment said they are answering two different questions.
@@reubensavage2067 She's always woken up, otherwise you couldn't ask her. Prepending the question with the pseudo-condition of her being woken up doesn't actually change anything because it always happens. The question is fully equivalent for "What's the chance heads came up?" which is clearly 50%.
I see what you are trying to do. You view each waking up event as an independent event and try to assign a probability to that event (just like Derek proposes in the video), but that approach is flawed since they are not independent. Monday Tails and Tuesday Tails cannot happen without the their also happening.
Imagine you have a somewhat unusual coin that instead of heads has one dot on one side and instead of tails it has two dots on the other side. Each dot represents a waking up "event". After the toss pick one of the dots you see (which is either just one, in which case the choice is simple, or two, in which case you just randomly pick one, since SB can't remember being woken up before, the order does not matter) and ask yourself "What is the chance I see this particular dot because the coin came up with the single dotted side?". If two dots were up you answer the same question for the other dot. The experiment is exactly equivalent (if you don't agree, please explain). Do you still think the answer is 1/3?
@@nekekaminger The part that I disagree with is that I'd argue they are independent events. She could be woken up on Monday Heads and be asked the question, or woken up on Monday Tails and be asked the question, or be woken up on Tuesday Tails and be asked a question.
As others have stated, it really comes down to which question is asked of her.
If she's asked "What do you believe is the probability that the coin came up heads?", then she should answer 1/2. Because the coin either came up heads, or it came up tails. It doesn't matter which day she woke up; the coin was either heads, or tails.
If the question is "What do you believe is the probability that you were awoken on heads?", then she should answer 1/3. Because as I mentioned in my first paragraph, if she's asked this question on Monday Heads, she would be right. If she's asked on Monday Tails, she would be wrong. If she's asked on Tuesday Tails, then she would again be wrong. So it's a 1/3 chance of her being right about the 2nd question.
@@ThrowAway-hy5sp you haven’t tackled his point that these events are not independent. Monday tails and Tuesday tails are essentially the same event. For the example where she wakes up “a million times” it’s 1/2 chance that she’ll wake up a million times or 1/2 chance she wakes up once. Either way if she wakes up on the thousandth Tuesday and is asked “what’s the chance that you will wake up another thousand or so days”, its 1/2 as is “what’s the chance you only wake up today on the monday”. There’s not “more chance” of waking up in the millionth day like it’s compared to being in a simulation. It would be like saying the chance of you living in reality is 1/2, and the chance of you living in any of the millions of situations is also 1/2. 1/3 would be the answer to “what’s the probability today Is Tuesday” regarding the original question.
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?"
This is a brilliant comment. Contrasts very well the difference that is muddled in the "what was the probability" question. The answer to, "what is the probability the coin was heads?" is objectively 50%. The answer to the question, "what is the probability that the coin was heads AND that your answer is correct?" is 1/3.
@@bbanks42 What? So for the first question your answer doesnt need to be correct for it be... correct?
The answer for "what is the probability of the coin to flip heads" is indeed 50%. But thats not the question, the question is "what is the probability the coin FLIPPED heads" with the given that if you are being asked that question you woke up. Similarly if someone flips a coin and it results in a Tail, it would be correct to say the probability of flipping tails in the past is still 50%, but wouldnt be correct to say the probability the coin that was flipped was tails was 50%, because you are clearly already seeing the result, and its 100%.
Imagine if the SB only woke up if the coin was Tails and was asked "what is the probability the coin FLIPPED heads?" , it would be ridiculous for her to say the chance is 50% after being asked that question, because she knows she wouldnt be asked that question if it was Heads.
Makes no sense. The setup is wrong. You cant make her forget that she woke up yesterday .... and then ask a logical realistic question.
If she cannot remember yesterday, then the asking person might have forgotten who they are altogether, or whom to ask.
Like : you are my banker, with a brilliant mind, and you can recall all of my bank statements from memory. But i always forget my adress, my name, my job and which bank to go to.
Now you want to ask me a meaningful logical game theory question on how to save money better ?
Makes no sense.
....also the coin is a half half deal. The monday or monday-tuesday thing is a scam. Please try Mon and then Mon Tue with a coin.
Heads comes up .... or both sides come up.
Great.
@@crockmans1386 go watch cartoon
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.
Exactly
So the case (Tuesday + tails) is absolutely useless to the probability of heads occuring.
As it was never defined that heads will result in her waking up in tuesday
I think the problem is she doesn’t know what day it is
@@Nanbread-bw7nq If the problem is dealing with what day it is that she was woken up vs. what is the probability that the coin landed on heads, then those are 2 independent probabilities.
The probability of the coin flip (reminder that it only occurs on Sunday) landing on heads is and always will be 50%
Now if we consider this specific problem and its wording, then there are only 2 days that she will be woken up: Monday and Tuesday. This information is what is always known (she doesn't forget the problem itself, only the outcomes she is told when she is woken up and then goes back to sleep).
Okay, so now we can ask the following question: What is the probability that today is Monday? well we know based on the information provided in the problem as it was worded, that she would only be woken up on Monday or Monday and Tuesday, which means we can eliminate Wednesday through Sunday. This means that the probability of the day she wakes being Monday is 50% and Tuesday is 50%...if she assumes it's Monday then the probability of the coin flip stands to be 50/50...if she assumes it's Tuesday then it's a 100% probability that the coin landed on Tails.
If the problem accounts for more days being woken up, then that changes the probability of what day it is that she is woken up, but that's a different question than "what is the probability that coin landed on [insert heads or tails]".
@@N8ive49er But the question is not asking what the probability of what a coin toss "will be". It's asking about probability of what it "was", given that you were already woken up after the experiment has started.
Wow it took until the example at the very end that I understood this. Went from a halfer to thirder
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 also feel like they just missing the obvious lol
If the question for the sleeping beauty was to tell if the coin flipped heads or tails. She is woken up three times,two times if it's tails and just one time if it's heads. If she says tails all three times, she'd be correct 2/3 times. If she says heads, she'll be correct 1/3 times. In conclusion, the probability of her getting the answer correct if the outcome is heads is 1/3. Whereas,the probability of the coin flipping to either heads or tails is 1/2. She would be right 1/3 times,but then answer is 1/2 as per the question.
@@architlal8594 the question doesn't ask her to predict whether the coin was heads or tails. it asks her what the probability is that it was heads.
so her response wouldn't be 'heads' or 'tails'.
the trick with this problem is that people are fooled into thinking that monday (tails) and tuesday (tails) are independent events. but they aren't. they're actually the same event. the reason you get the 1/3 distribution is that she gets woken up twice on a tails. and therefore gets asked twice from the same coin flip.
I feel like that arguing with people about politics and society all the time. In the absence of an obvious answer on a lot of those issues - unless you are very well informed on them, which a lot of people aren't, often times people just try to roll you with fallacies like that. I believe usually even unknowingly so and thinking "they got you". But it's very tough to effectively counter that, especially in the moment, because, as this video shows, unraveling such fallacies can be very hard. Often much harder than coming up with them.
Actually a lot people that understoodd this question as "did I flip a heads or tails?" respond with "I dont know its 50/50." This isnt some word game this is a sort of paradox. The people that disliked this video isnt arguing that a coinflip is always 50/50.
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 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 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.
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"
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
Conclusion: Odds and Probabilities are very different things.
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.
I feel like no matter how many times they wake you up, it still was 50/50 whether you're gonna sleep one day or million
Correct.
It is a 50/50 chance on sleeping 1 day or a million.
However, each time she wakes up its a 1/1000000 chance of being tails
@@joshpollnitz1618 Each time she wakes up it's 50/50 chance she either exists in the heads sequence or the tails sequence.
The sequence is one complete package. It doesn't matter how long the sequence is, it's still one package. The reason being that the conditional statement is the fair coin which only has two outcomes at 50% chance each.
You are confusing the probability of the coin coming up heads _before_ it was flipped with the probability that it _did_ comes up heads, after it was flipped and the observer has some amount of information pertaining to what the outcome was. These are two very different things. The answer to the first is always 50%, but the answer to the second is not necessarily 50%, depending on what information the observer has observed. This is the fundamental principle behind Bayesian probability.
@@therainman7777 Sigh.
Leftism is a religion upon which the idea that an individual's perceptions can dictate reality is the central object of divine worship, thus the solipsistic imposition of the self's mind onto the world makes the Leftist a god in his own eyes.
Pay attention to me. There is no secondary set of probabilities. There exists no such thing as metaphysical probability. That's a perceptual lie that the illogical parts of the human brain concoct in order to alleviate the pain and stress caused by the sheer gravity of reality pressing on you from all sides forcing you to accept that you cannot escape objective truth.
There is no such thing as "probability after the fact" nor is there any such thing as a "choose your own adventure" kind of probability. Probability is determined by material scientific forces that can be measured and predicted to a certain level of accuracy. In this case though we have from the outset a theoretical "perfect" object in the form of the fair coin. We know from the get go what the probabilities are because those probabilities are deterministic and they are determined by a physical object whose attributes are already established -- the coin is an ideal one.
According to the parameters set by the use of such an ideal instrument we are bound to make the conclusion IMPOSED by the parameters set by the ideal instrument. You cannot escape the physical imposition of physical laws even when those laws are purely theoretical because those laws still follow the rules of logic. The coin is set to have a 50/50 chance to land either heads or tails therefore those are THE ONLY outcomes you can measure in the experiment. Nothing else besides those two outcomes exist.
Period. End of story. --
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”
It will scares me if one day this happens in real life
what is this man saying? is he from another universe?
I got a bounty commercial lmao and I also posted a comment about it before I saw yours😂
@@mgkelley2609 😂❤️
Well that's a nice dream!
This is a scoring problem.
If you ask after every waking and record every correct guess as a point, you're assigning two points to tails and one point to heads because you asked more often on a tails. You're no longer answering the original question.
On the marble variant, you'd need to split the marbles into two bags (a white marble in one, however many black marbles in the other) to equivocate. The question is still "what are the odds of drawing a white marble?" but you present the first bag on heads and the second on tails. There can be a million black marbles and the odds are still 1:1.
This problem is in real life in managerial accountancy.
Specifically in risk calculation. Everything has a risk and some riskier business has often a bigger income. The risk is converted into money by multiplying the chance of failure by the cost of failure (and similarly the success is degraded by it's chance) you subtract such value of risk from the value of success and you get the value of this particular business decision.
As far as I heard the accountants say that the chance is always the same no matter how many times you take it, which day of the week it is and what happened in the past.
I guess they wouldn't agree with the thirders.
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
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
This seems like the same as the game show dilemma where you pick one of three doors to find the prize behind one, only the game show host then eliminates one door where the prize is not (never the door you've already chosen) and asks if you'd like to now switch to the other remaining door.
In this case, it's the probability of the day vs the coin.
Twice as many days for tails -- twice as likely she was asked this question on a tail flip.
I have thought about this for several days and I can reason both answers; however, I think many are thinking about this from her point of view. As to when she is awaken, it is always the first time and thus 1/2 will always be her answer. When perceived from our view point, the answer is 1/3.
When he said "Don't hit the like / dislike button" , exactly at the same time my like and dislike icons in UA-cam started "Glowing" .....what is that ? Magic?
AI when it hear like it glows
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.
In case of marble in the box would be true to assume (relative to coin) that you have 2 boxes: first with one true marble, and second with any quantity of false marbles
Choosing true marble is still 50/50
The difference here is between objective and subjective. The coin flip is the object and Sleeping Beauty or her model equivalent is the subject.
As object is as it is regardless of subject for such an event, one ought to take the probability for the object and not the subject since the subject's state is contingent on the outcome of the object. Since the object is causally upstream of the subject, it is the load-bearing event for the overall dynamic in question. The resultant outcome is a contingently dependent but different action.
Therefore it is 1/2.
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 😎
Stolen from Tom Scott. Doesn't bother me, but it is.
UA-cam algo like that? It might get changed soon if it can be abused.
@@theeraphatsunthornwit6266 Pretty much all social media is optimised for controversy or moral outrage, because that's what drives the most interaction. I don't honestly have any idea about the YT algorithm, but we can be pretty sure it'll rank videos with a widely split opinion above a video that has just a high number of likes, especially if there are lots of comments too.
@@bingbongthegong what video did Tom Scott do that abused the like count?
@@bingbongthegongThat would be the like and dislike number update
The ideo of ratios was not stolen though
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!” ??
Good analysis!
exactly my answer. perspective and probability are two different things. and counting one event twice, as he did in the experiment when he got 1/3rd each does hurt people who do statistics.
In this case ‘probability’ refers to her ‘credence’ of the coin being heads, i.e. her subjective confidence all things considered that the coin was heads.
That she would have slept through Monday on a tails flip is completely irrelevant as her credence of each scenario is not equal. It would only be rational to assign a higher probability to tails if her waking up eliminated some possibilities of heads which it doesn’t, so the chance of her being in a heads-world is exactly the same as being in a tails-world.
Your doctor case is not analogous, since waking up eliminates all possibilities in which you die, so you gain information from the fact you wake up.
Sleeping Beauty actually does gain information from her awakening: It's no longer Sunday! It's either Mon- or Tuesday. On Sunday the coin is flipped: 50% chance for heads or tails. SB's Sunday credence is intact. Now she is awakened: Oops! Is it Monday? Is it Tuesday? She knows it not. Her presence depends on her past. Of course, her memory of the Sunday chance seems intact: 50% for heads or tails. But her memory of that past lies in the very presence which depends on it: a loop--not to be trusted! Ask her this question now: "What do you believe, my dear SB, is your chance of being awakened again?" Hm..., ...Monday 50%, ...Tuesday 0%. How to answer? She's no longer in a "Sunday mood". SB's credence has been compromised by the fact that her beautiful presence may have been tossed into a "Tuesdayish" tails-tails-nightmare already. The very bed she sleeps on has been gambled with. There's a chance she lies on a doomed bed (at least until the Prince of Mathematics appears on a white horse; allow me to cry for a while--but only with one eye).
The SB-problem wants to not just entail the tossing of the coin but the tossing of the tossing itself. SB has already been tossed and turned in her bed (lousy sleep?) before she awakes. Her answer doesn't come from a 100% heads- or tails-world.
Btw., to address another point in the video, I think, it doesn't matter if she'd be awakened 1.000.000 times with an original tails flip: It's Monday versus 999.999 days presenting Tuesday. The tossing has only been tossed once--not a million times.
Are the odds skewed for tails tough? When it's heads she won't actually wake up on tuesday.
It's a variation on Pascal's wager.
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.
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.
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.
I agree that it's 1/2. Even if it was 1 vs infinity I would say 50%
That makes the most sense to me as well
A branching reality is the best way to describe it. You don't care about the probabilities in one branch, only the probability of ending up in one branch or the other.
I think you misunderstood the question. If you wake up a million times more often if it hits tails, then it is a million times more likely that when you wake up, it was tails.
@@johnnysilverhand1733 1:04 how the question isn't what the likelihood of a coin coming up heads or tails ??
such an awesome "feature" that youtube not only removed the surveys in videos, but also the dislike count 😍😍
The number is still being transmitted with the page, but not being displayed. If you want to see the numbers again, you can use a browser extension such as "Return UA-cam Dislike". This is why I'm able to tell you that at the moment, this video has 114k likes and 61k dislikes.
@@mattgies unfortunately that does not work for mobile. and that extension is not accurate, it is only a rough estimate :/
@@mattgies The UA-cam API doesn't include the Dislike Dataset no more. The Extension just looks at every Extension user which likes or dislikes the video and extrapolate the data to get a rough estimation of the like to dislike ratio
@@therealdnold Oh wow! I had no idea
@@anonymouscommentator Youtubе Vanced on mobile does include Return UA-cam Dislike as a toggleable setting, so it is still possible to use. However, that estimation is still a factor.
i have came back to this video after a year. and now im having a debate against myself because i choose 1/2 and now my answer is 1/3
so your memory wasn't erased? :)
Here’s how I’d argue for the thirder position. In the initial probability distribution there are four possible scenarios she could find herself in. Each with a 25% chance. Since her memory is erased, there is a 50/50 probability over the day of the week in addition to the 50/50 coin outcome, from her perspective. When she wakes up, Bayesian collapse occurs on the Tuesday sleeping state, which gives her new information. The lost 25% is redistributed between the remaining cases, moving the final probability to 1/3 after the new information comes in.
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.
Having notifs on made me saw 0 agrees 0 disagrees and I thought this was gonna be a video about how you can never have everyone agree on anything XD
I know, I was confused too
Same
Outstanding explanation. It depends on how you understand/perceive/interpret the question.
More like on whether or not you understand what is being asked. Because it's not really open to interpretation. :)
1/3 should show us that perception and reality don’t have to be the same. We can reason through counterfactuals but can’t live in them.
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.
But then you would disregard the information she already has, and information is important in mathematics.
If she gets woken up, theres a 1/3 chance that its tuesday.
@@gorgit That's not the question. First, she has no information other than the rules of the game. The question is "what does she believe the probability is that the coin landed on heads?" When she gets up, she isn't given anything other than the question. 50%... problem solved.
@@JohnJJSchmidt she is given knowledge that one of the days she could be awoken is Tuesday, but she doesn't have the ability to tell what day it is when she is awake. Therefore she has the knowledge that it could either be Monday after a heads, Monday after tails or Tuesday after tails. Even though the question she is asked doesn't change the probability the pre existing knowledge she has does.
@@jaredhahn7970 The probability of what? Again. The question is what does SHE THINK the prob the coin landed heads, not the prob of it being a certain day. If she says anything other than 50% with a blank mind, she is letting the pre-existing knowledge of the rules interfere with really basic reasoning.
If we phrase it as “what is the probability that she woke up in a scenario that the coin was heads?”, it’s 1/3; but yeah, given that the question is, “which way did the coin come up?”, it’s still just 1/2.
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.
Just image a portal, which could lead you to three different places, an ocean, a continent and another continent. What is the probability of getting an ocean? One-third.
Similarly, the question didn't mean what is the probability of getting a head or tail but rather, what day do you think you woke up, the Heads Monday (Ocean), the Tails Monday (First Continent) or the Tails Tuesday (Second Continent)?
It is also similar to counting the probability of getting an even number among a box containing 1,2 and 4, which would be two-third. Here, we don't need to find the probability of selecting a random number and it being even (which would be one-half), but rather choosing it from an already predictable outcomes (i.e, 1,2 and 4).
So, in a nutshell, the answer would be One-Third.
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P.s. After typing all these, I think the reason it remained unsolved is because of a group of children quarrelling over it. And when that happens, it doesn't matter what's the right one. 😅
But your initial analogy is wrong, and asking which day she thinks it is is a red herring. The monday and Tuesday of tails are not multiple chances of tails being right, they are instead a chain link of smaller possibilities. Let me explain.
Let's say I'm going to send you money. There's a 50% chance I'll send it in the morning, 50% I'll send it in the afternoon. If I send it in the morning, it'll arrive on wednesday at 12 pm, but if I send it in the afternoon, it'll arrive on Thursday morning at either 8am, 9am, 10am, or 11 am.
A few days later, the money arrives, and you realise you don't know what day or time it currently is. What are the odds that the time is 10am on Thursday?
It's 1/8. Not 1/5. The Thursday times make up 50% of the probability collectively, and the other 50% is made up of the Wednesday chance.
Just because there are more days of awakening on the tails side doesn't mean that they carry the same probability as the solitary awakening on the heads side. The tails instances don't build upon each other. At any one moment of being woken up, there was a 50/50 chance that the coin was heads or tails, and that doesn't change despite being asked potentially multiple times.
@@TheShepdawg9 It feels kinda right as well now that I think about, but I have Mathematics exam day after tomorrow, so I don't really wanna spend any time countering your example.
Sorry. 😅😁
I love this kind of stuff. Looking at different points of views is fun for me.
My initial reaction was 50%. At the same time, you can't argue with the real test that shows its 1/3. But then again without knowing what day it was, maybe you should still say it's 50%.
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."
Of course. The statement as 1:53 is simply false. These are two different questions, each yielding a different probability distribution and thus different answers.
It's two levels of abstraction using the same symbol so the English confuses the math.
Lets do the same exercise but change the coin to marbles when we present the new abstraction instead of hiding it behind the same name (coin flip)
Flip a coin every time it turns HEAD place a RED marble in a bag
Every time its face up Tails put two BLUE marbles in the bag
Now if we ask the question "What are the odds of the coin" well its 50/50
What are the odds of pulling a red marble out of the bag? Well 1/3
Paradoxes are cool, this isn't one, just a poorly worded question
Yeah this was my thought too. However, it does ask "...IS the probably that the coin CAME UP..." So it is not asking you how often the coin did anything. It is asking how often you will wake up because of tails as opposed to heads, therefore it must indeed be 1/3rd.
@@identifiesas65.wheresmyche95 There is no probability for events that have already occurred. Hence, the question if phrased that way is about whether you *believe* it did or not, and that belief is where the probability component enters the picture. Whether Bayesian or frequentist, you'll be thinking about two things: "What is my belief that a fair coin would have already been heads or tails?" (naturally, ½), or, "What is my belief that this awakening is a heads awakening?" (naturally, ⅓).
If we are clear about which question we are asking, the problem goes away.
Edit: I think it's actually really useful to treat this as one would the Monty Hall problem. There, it becomes a lot more clear what's going on if you presume a hundred doors, or a thousand, or the like. If you pick one door out of a thousand, and Monty opens *every single other door except one,* would you switch? It seems pretty clear you should. You only had a 0.1% chance to pick the right door at first. Monty has now eliminated every other door *except one.* The odds are enormously in favor of that other door. It just happens to be hardest to intuitively see that when you have the smallest possible number of doors (3, in this case.)
We see the same thing with the SBP. We only have three awakenings (well, one vs two). What happens if we make it one vs 999? Further, what if we add some expected value to the answer?
Consider: Sleeping Beauty wins $1000 if she correctly picks Heads, and $1 if she correctly picks Tails. The expected value now depends on how you view the question! If we structure things on a *per trial* basis, then half the time the coin is heads, and half the time it is tails (before any awakenings have occurred), this is agreed by all parties. Hence, *per trial,* the expected value is $1000 if she guesses heads correctly, and $999 if she guesses tails correctly. Since each is equally likely *before* any awakenings have occurred, she should choose heads every time; she will net more money, albeit slowly.
If, however, we award her the exact same prize for any correct guess on each awakening (e.g. "if you pick a side of the coin and are correct, you win $1"), then she should 100% always choose tails, because she can win $1 on half of trials, or $999 on half of trials. The preponderance of *awakenings* is on tails paths.
Someone asserting that the probability must be ⅓ is claiming that, for _all_ experimental setups, the higher expected value for these Sleeping Beauty prizes must be from picking tails. This is not true. By offering prizes based only on the coin's facing, *not* on the number of times Beauty awakens, we can clearly see the difference between the two approaches.
@@identifiesas65.wheresmyche95 - Does she even know about the multiple awakenings? It's not made clear in this video (and it's my only familiarity with the problem).
As soon as you mentioned a "less dominant soccer team," I immediately thought, 'Oh he's gonna use us isnt he' lol
"In this instance of waking: how likely is it that this specific instance of waking is associated with the outcome of heads?"
The problem with some of the scenerios is that we don’t know if the coin flip took place. But it’s rational to think that if you are woken up there’s more scenarios that exist if it’s tails so even if the odds of the coin is 50/50 you’re more likely to exist in a scenario where tails was flipped.
I feel really bad leaving a Veritasium video on "dislike", I even almost liked the video as a habit after I was done watching it.
which also leaves me to believe the like dislike ratio is not to be trusted that much.
Yeah I feel like the number of likes is mostly a combination of people using the like button to actually "like" the video and others who only listened to the first part. The comments seem pretty consistently to agree that it's 50/50.
The ratio in the title would also cause a lot of people’s opinions to change due to peer pressure.
I think he designed it so that he'll receive more likes than dislikes since statistically, more people will choose the 1/3 option. I'm pretty sure he thought through it.
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).
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It's this bloody simple. Counting tails twice doesn't change the chance.
Yep. Think about the marble example. Its not pulling one marble out of a bag of 1 million black marbles and one white marble.
It should be stated - flip a fair coin then if its heads pull a marble from a bag of 1 white marble and if its tails pull a marble from a bag of 1 million black marbles. Whats the probability of a white marble? 50%
@@rakino4418 close. If it's tails, grab all the black marbles. Now you have a lot of black marbles and only one white marble.
The chance to end up with that many black marbles is still 50%, but that analogy more closely resembles the mess that is this probability discussion :')
Exactly, what that test is saying is that 1 heads = 1 heads, while 1 tails = 2 tails. The latter is then saying 1 = 2, which cannot be.
This is clearly an example of a sampling problem. The probability of the sample changes from the priors because the results are unequally sampled. So it matters very much what specific question you ask her: what is the probability of "a" coin coming up heads? vs what is the probability of "your" coin being heads? If she knows the rules of the game _before_ falling asleep, then she knows she will have a sampling bias for tails. If she does not have that information, than she would pick 50/50.
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.