Brady hit the nail on the head, "What is the probability the coin was heads?" is a slightly different question from the question "What is the probability that sleeping beauty is woken and the coin was heads?", which is the question that you're always actually asking sleeping beauty, since she has to be awake to ask her.
The catch is that the question is NOT what YOU think about the probability that the coin was heads, but what sleeping beauty thinks about that probability when see is awake. That's why it's a paradox, got indeed a bit confusing as the interview went on.
Asking multiple times without re-flipping the coin doesn't change the probability of flipping the coin, but it changes your probability of getting the answer right.
@@giladkay3761 It also depends on exactly the question being asked. "What is the probability the coin came up heads" is obviously 50/50. "What is the probability that we woke you up to ask that question" is not 50/50. It's entirely possible for me to have a fair coin and tell you "I will ask you how it came up only if it comes up tails." That doesn't change the probability of the coin flip, but it totally changes how you should answer "what do you think came up?"
@@darrennew8211 I think that is the confusing part. Because we assume the questions mean different things by the fact that different probabilities came out for each question, instead of logically and semantically defining the difference between them.
It doesn't change anything. The waking up occurs AFTER the coin toss. Saying anything that can happen after the coin toss has any bearing on the toss itself. violates the law of causation, because toss causes waking up.
The thirder and halfer arguments are talking about two completely different probabilities: the probability of the coin being heads *given* that Sleeping Beauty was woken up, and the probability of the coin just being heads. One is conditional, the other is not. Which goes back to what Tom was saying about it being about what we're really asking Sleeping Beauty. Are we asking about the conditional probability or the unconditional probability?
Exactly my thought, and I think that Tom saying that both views are supported by mathematicians is a little misleading. The real lesson shouldn't be that statistics depends on your opinion, but that statistics depends on the facts that you are given.
@@ΠαναγιώτηςΓιόφτσος Yes, I honestly take offense to the statement that both are supported. One view is correct, the other view is wrong. There is no gray area here.
@@ΠαναγιώτηςΓιόφτσοςI disagree slightly on the last part, statistics depend on your interpretation of the question and the information you are given. The same information and question can be provided, but interpreted differently with different results. And that is often used to twist statistics to show what the person wants them to show.
Well it was more likely an outcome, than the thirder option being factually correct, after all no matter how infinitesimal the outcome of you being the sleeping beauty in a Numberphile video may be, let's say 1/TREE(100^10^3.1) that's still infinitely more likely than the thirder option which has 0 probability of being correct. A coin was flipped, it had 50% chance of being heads, no matter how many times anyone wakes up to answer the question in the first place.
@@livedandletdie Of course the coin comes up heads 50% of the time but think of it like this. Every time it comes up heads you are shown the coin once, every time it comes up tails you are shown it twice. The coin is still fair but picking a random viewing of the coin, what is the chance you see a tail?
Like Brady was getting at, it's not a mathematical paradox, but a language paradox because the question is vague enough that it can be interpreted as asking a simple question of the probability of a coin flip, or as asking the likelihood of waking up by a heads or tails. (There was a 66.7% chance you were woken up by a tail, but objectively only a 50% chance that a tail was flipped. Those are two different answers, assuming two different interpretations of the initial question, that don't actually contradict each other)
Except it is a mathematical problem, and that is that the precise question matters. Mathematics without context is useless. Mathematics is only useful because it relates to problems. In this case how situations influence otherwise fixed probabilities
There is actually a 50% chance that you were woken up by a tail. And if it was tails, then there is a 50% chance it is Monday and a 50% change it is Tuesday. If it was heads, then there is a 100% chance it is Monday. So in total: Monday+Heads 50%, Monday+Tails 25%, Tuesday+Tail 25%. That in the case of Monday+Tails you will also be woken up the next day makes no difference here. That in the case of Tuesday+Tails you were also woken up the previous day makes no difference either.
@@ronald3836 lol, that doesn't make any sense. There's a 66% chance you were woken up by a tails. That part is out of the question. It's because if it's tails, you'll be woken up twice as often as when it's heads. It does matter that you'll be woken up twice for a tails, it matters very much. The "paradox" is in asking "is the coinflip heads" vs "you were woken up: is it heads?". By rephrasing it you have solved the paradox and are now just getting the wrong answer :")
@@JoQeZzZ in the whole experiment, there is a 50% chance that you are woken once by a heads and there is a 50% chance that you are woken twice by a tails. This follows directly from the coin being fair and the description of the experiment. In the tails case, you wake up twice. For one particular awakening, the chance of waking up on Monday is half the probability of the tails case, and the chance of waking up on Tuesday is the other half. So both have probability 25%.
yea I think on high levels of math like this, real paradoxes can't really exist anymore. There's Law of Excluded Middle stuff if you go into the depths of logic, but with things like this there's usually a sleight of hand or a subtle misunderstanding somewhere. If you've seen jan misali's video on paradoxes, he discusses how there are REAL logical paradoxes and linguistic/trickery/counterintuitivity paradoxes, and that's kinda the distinction I mean here.
I don’t get the thirder position tbh. Let’s even assume that one gets woken up 1000 or a million times when it’s T and only once when it’s H. If I’m sleeping beauty and I get woken up and asked the question it seems like it’s 50% that I’m on the streak and I don’t remember (and will not remember) the other times of the streak and 50% chance this is the only day I’m asked. (Although if I’m betting it’s obvious it’s better to bet on T assuming one wins an equal amount each time one is woken up. But I guess this changes if one maybe looses some proportional value each time one is wrong)
@@Nia-zq5jl Thinking in terms of betting is exactly the right way to reason about this. From SB's perspective, tails will be the winning bet 2/3 of the time. So 1/3 chance of heads is the right answer.
@@jonathanmooser6933 The halfer response to "If you are right we will give you 1000 pounds" would be "If you are right EVERY TIME WE WAKE YOU UP we will give you a thousand pounds at the end of the experiment". Perhaps a middle ground offer would be "at the end of the experiment we will give you a thousand pounds divided by the number of times you are woken then multiplied by the number of correct answers you give". That way there are consequences for bad bets. I think this offer really leans towards the halfer worldview though because if you systematically chose heads or tails you would get the whole money half the time and no money half the time. Even using probabilistic choices, flipping a coin each time you woke up to choose would lead to £500 on average. With a 6 sided dice we say tails on 1 though 4 or heads on 5 or 6. If it is tails we expect to win 2/3 of the money and heads we expect to win the money a third of the time. This leads to the expected winnings overall to be £500. Ultimately the "winning 2/3 of the time vs winning 1/3 of the time" dilemma is more akin to "I'm going to flip a coin. If I flip tails and you guess it you get £2000, If I flip heads and you guess it you get £1000, if you guess wrong you get nothing. What do you think it will be?" The probability of the tails isn't higher but the expected winnings of guessing tails every time is twice as big as guessing heads every time
At least it's better than those thought experiments about burning cats or putting babies in blenders. I'm not sure consent would help with tying people to a railroad track, though. XD
The third argument is the probability per day. The half argument is the probability per experiment. They're measuring different probabilities. Imagine the experiment is done 10 times, half the time it shows up as a head and half the time it shows up as a tail. And she will always guess a head. Well she will be correct in 10 days out of 30, but she will be correct in 5 experiments out of 10.
With the betting version, if she gets a payout every day, then the thirder stratergy works. If each day she is asked whether she wants to commit to a single bet when she is finally woken up, the halfer stratergy works.
Just consider the scenario where she's only woken up if it is a tail and not at all if it is a head. In that case, if she is being asked the question at all, then it means 100% that the experimenter has flipped a tail. Clearly the 1/3 answer is a posterior probability that is completely arbitrary and determined by experiment design. It is the probability that heads HAVE BEEN flipped, like the probability that it HAS rained if you see puddles on the street, which does not tell you anything about how likely it will rain at any given time.
Yes I had the same thought. It is kind of like the question, what is the probability that our universe supports human life. People may argue the probability is infinitesimal. But the only universe in which we can ask the question is one which supports human life. So the answer has to be one.
Seeing puddles in the street increases the likelihood that it rains and acts as evidence. Waking up does not increase the likelihood of tails and acts as zero evidence. Thus the posterior equals the prior.
I feel that there there are two 'probability spaces', one from an external observer and one from sleeping beauty, and these two are squished together. The events from the frame of reference of sleeping beauty is squashed into one half of the outside observer, and the outside observer's space is squashed into the 1/3rd of sleeping beauty.
I've a strong intuition this lesser-known paradox is tied to two infamous ones: the "Monty Hall problem", and the seemingly impossible "observer effect".
@@donweatherwax9318 Monty Hall isn't a paradox though, there is one and only one correct answer to monty-hall questions. This Sleeping Beauty Paradox has two valid answers,i.e. a true paradox proper.
@gregoryfenn1462 Granted; however, I'm not sure _this_ one is really a paradox. 'Paradoxes' have been solved, as Monty Hall has been; and I feel like this one may have a correct answer too. (And just as with Monty Hall, perhaps not the 'intuitive' one.)
The confusion arises when you think of every “wake up” being independent, but they aren’t. The 99 wake ups on T is 1 event and the 1 wake up on H is 1 event. There are only 2 events, not 100 events. The answer is 50/50 and the princess gains no more likelihood of being right for saying T. From her pov she only wakes up once and only has one guess.
"If you guess correctly, you gain $1." Clearly, this would lead you to guess T because the expected value is higher. But, really, that's just a different way of saying "If you guess T and are correct, you get $99, and if you guess H and are correct, you get $1."
@@renmaddox Now try other two variants: 1) If you guess wrong you loose $1 and if you guess correctly you gain nothing. 2) If you guess correctly you get $1 and if you guess wrong you loose 1$.
@@kgsws 1) Guess T and are wrong lose $1, guess H and are wrong lose $99. Still guess T 2) Is just the individual cases for gain/lose if you stick to the same answer each time. T and correct get $99, T and incorrect lose $1, H and correct get $1, H and incorrect lose $99
@@kgsws The first of those doesn't really change anything meaningfully. It reverses the "right" choice, but for the same reasons. The second is seems to be effectively the same as my version: Might as well guess H, because either you're correct and win $99, or you're wrong and lose $1.
You can make it more obvious by making two different games: 1) Give her 1 point each time she guesses correctly. This results in the thirder position. 2) Give her 1 point only if she answers correctly every time (note that since she has no recollection of having woken up before she will necessarily guess the same every time). This results in the halfer position.
You get woken up NOW. You're asked to give your best guess of the coin flip NOW. It seems quite unambiguous that the imaginary point will be given to you based on your answer NOW, and the value of your answer won't be halved if the coin landed on Tails.
@@eugenehertz5791 terrible analysis. This is a new scenario with more detail added by Ben - it's "quite unambiguous" that this scenario wasn't presented in the original problem and there were no points given at all. Either get on board with the updated situation or ignore it, don't try and force your own third completely new situation and claim it's the "true one"
I disagree. The first game will lead towards the best strategy being choosing at random. In the only monday case, you have a 50/50 chance of saying the right response, and in the multiples case, you on average will get half of your answers right. The second game, best strategy is always say the same thing. (50/50 chance of being right.)
Asking her the question “is it heads or tails?” is different than asking “is her total score for correct vs. incorrect higher if she always answers heads or tails?” Then you realize obviously what she should say.
I think this is the closest insight in the comments. The problem has nothing to do with conditioning on "woken" but something to do with " the probability that a YES OR NO question being answered correctly" vs. "the probability of a coin flip".
Your explanation was FAR better than Veritasium's, I finally understand what's the paradoxical part about this. It's a fun sleight of hand! Of course the probability that the coin comes up heads remains 1/2 no matter what, but if that's true then why does the calculation say 1/3?? Because the assumption that these three equal probabilities have to add up to 1 is false! These are just the cases where SB is woken up, which is 3/4 cases, so that's what the probabilities add up to. If you add up the case where the coin came up heads on a tuesday (which means SB is not woken up), THEN you get to 1. And yes it's the same as the other three again, so we get 1/4 for all of them, and properly 1/2 for all the Heads cases combined. So indeed the difference is if we're asking about the coin's probability distribution in general (which never changes from P(H) = 1/2), or about what the coin came up as THIS TIME. Which, because SB is awake, is conditioned on the fact that SB was woken up, and the answer is therefore P(H | SB woken up) = 1/3.
@@MrDannyDetail honestly agreed, I wish this explanation was in the video rather than in a comment by someone else. The part I meant that was better here, is just the fact that I'm taking the thirder position seriously now. I don't remember exactly, but at the time I don't think Veritasium did the thing of "proving that all three wake events have the same probability"... he probably just said it, and my response was "wait wtf no they don't!" -because they don't unless you do the "conditioning on SB waking up" trick- ok they do, that was actually a mistake I made at the time. But their probability without the conditioning trick would be 1/4 and not 1/3, so the point still stands. Anyway, the fact that the calculation was done here and naturally gave 1/3 blew my mind. And now that I understand how it's actually a valid calculation that answers a sensible question, that has really expanded my perspective on this.
The question asks about what event happened when the coin was flipped (t or h). When the question is repeated on tuesday it doesnt change that there was only one event that could lead to that. So when they list out P(mon n T) and P(tue n T) they are equal because they are the probability of the same event. Listing both out double counts them and if you count them just once the paradox goes away.
I think there could be two interpretation to the problem: 1. The probability of coin flip being a tail. 2. The probability of you being correct if you answer tail. These are two different questions and the probabilities are different.(to be honest I feel like this is an ill-defined question)
I agree with Brady. You do different things AFTER the coin was either a tail (50% chance) or a head (50% chance). What you are asking isn't the probability the coin would land on tail or head, but what is the probability of a COMBINATION of things (one thing being tossing the coin, and the other thing being the different amount of time you will then be awaken). As Brady said, she is given more chance to be correct when it's a tail.
Back in college I did a project about the Brier score, which can explain the probabilities used in weather forecasting as an example. If we ask Sleeping Beauty the question every time we wake her up, she would minimize her Brier score by saying 1/3. If we ask her the question once during the experiment, it is minimized with 1/2. Since the question as stated was that "one of the questions she is asked is...", I'm more on the 1/3 side, but strongly understand the argument for 1/2!
There is one thing that wasn't mentioned which is, imo, the deciding piece of information. When sleeping beauty is asked the question, there is one piece of information that she knows that the amnesia drug can't take away from her: the fact that she has been woken up. Overall, there is a 1 in 4 chance that each of the following options will happen: It's Monday, the coin is heads, and she has been woken up; It's Tuesday, the coin is heads, and she stays asleep; It's Monday, the coin is tails, and she has been woken up; and it's Tuesday, the coin is tails, and she has been woken up. All of those options are equally likely and the coin flip is 50/50. But given that she has been woken up, that eliminates the option that it is Tuesday and the coin is heads (because she would still be asleep). The remaining three options are still equally likely. Therefore, given that she has been awakened, there is a 2/3rds chance that the coin was tails and a 1/3rd chance the coin was heads. This actually reminded me a lot of the Monty Hall problem and how 2/3rds of the time if you switch, you'll get the prize. Although the math is different.
"All those options are equally likely" is what's in doubt. If the coin landed on Heads, then her wake up must be the Monday, so 50% chance of Head+Monday. If the coin landed on Tails, then the wake up could be Monday or Tuesday, each with equal probability. So there's 25% chance of Tails+Monday and 25% chance of Tails+Tuesday.
The question is the probability of the flip, not a request for Sleeping Beauty to guess what the result of the coin flip was more often or some other similar question (such as what day is it). As stated in the video, the situation (and available information) never actually changes as it does in the Monty Hall problem.
There is actually an infinite number of events that aren't part of the experiment that you can't count. You can't include heads on a Tuesday in your math because it isn't a scenario that can happen and everyone knows it.
This reminds me of the anthropic principle. Our universe may be unlikely not by random chance but because in a likely universe nothing would be able to observe it.
Actually the sleeping beauty problem was invented by Arnold Zuboff and named by Adam Elga. The original formulation of the sleeping beauty problem was also about the antrophic principe. The question is whether or not your experience is itself evidence for something.
Brady's wagering argument is, I think, the key to this whole problem, or at least it's what unlocked it for me: What's changing isn't the probability, it's the odds. If you asked her to bet a dollar on the result of the coin every time she was woken up, then if she always guesses heads she's risking $2 to win $1, whereas if she always guesses tails, she's risking $1 to win $2. That makes guessing tails the dominant strategy, even if the coin itself is a fair 50/50 shot. The 1/3rd argument, then, isn't really asking how likely the coin was to be heads, it's asking you for the payout you'd expect if you assumed it was.
The two different answers are optimising different outcomes. 1/2 is the answer to give if you want to minimise the average number of times you are wrong _this one time the question was asked_ . n/(n+1) is the answer if you want to minimise, on average, how many times you were wrong _over the duration of the whole experiment_ . In both cases the average is over many copies of the whole experiment done iid. The two are subtly different, which shows that probability (because it's a degree of belief) is not only subjective but context dependent.
There is no optimising to be done. The is also no "being wrong", the answer is always the same, since the probability of the coin toss does not depend on how many times she wakes and thus is a constant number: 1/2
Exactly what I was thinking. The seeming paradox derives from ambiguity in language of the question posed to the sleeper such that they could answer in a manner that focuses on their true in-the-moment state of evidence (“halfer mindset”), or they could answer in a manner that employs a decision theory framework in combination with current evidence state. As usually formulated (including in the video) the cost/benefit matrix for use in the decision theory framework is itself left nebulous, but humans tend to like the feeling of being “correct”, hence the strategic dominance of a guess-tails strategy manifest by the “thirder” mindset.
I have seen this "story" before, and so long as the question is "what is the probability that the coin flip was tails", I have the same answer now: the probability is 50%, no doubt about it. Waking up one time or two times doesn't play any part in the probability because the two events are based on the same coin flip, whether or not she will be woken up in the future, or was woken up in the past plays no role in the probability. If the question was instead, "what is the probability that you have been woken up today because the coin flip was tails" then the question changes, and the answer is 1/3rd.
Ultimately this is all semantics and you've already stated the "intellectually honest" way to ask the question, but the simple inclusion of past tense in the way the question is asked is enough to change those semantics, at least in my opinion. The experiment is fully explained to you beforehand, you are aware that a coin flip happened and the result has been recorded. In effect, simply being awake indicates one of two things: It is not a Tuesday, or the coin came up Tails. In other words, 1/4th of the time when you are 'not woken up' is functionally the same as being woken up and shown that the coin came up Heads. If you were asked "what is the probability that the coin flip was tails" while actively being shown that the result was Heads, surely your answer would be "0%"? In the end this whole thing just seems like a clever way of restating the Monty Hall problem. By being awake, you are basically being shown what's behind "door number 3".
@@giladkay3761 if you wake up, there's a chance that it's Tuesday, and the probability that the coin flip was tails is zero in that case. If you're asked about "waking today in particular" the answer changes compared to being asked about "waking at any time"
@@Hyatice no, in this problem you are NOT given any information by waking up. You knew already that you would wake up, whether coins or tails. Therefore nothing changes and the probabilities after waking up are still 1/2 and 1/2. If heads, then you will wake up on Monday. This is clear. Half the time you run the experiment this will happen. If tails, which happens half the time, then you wake up twice, but both of those times you are not aware of the other time. So you can't tell if this is the first time or the second time you wake up. Both are equally possible. Probability that you NOW woke up on Monday is 1/2x1/2 and that you NOW woke up on Tuesday is 1/2x1/2. So if you wake up, the probabilities are: P:(H on M) = 1/2, P(T on M) = 1/4, P(T on T) = 1/4. P(H) = P(T) = 1/2 P(Mon) = 3/4, P(Tue)=1/4
It's either going to be a Monday or it's going to be a Tuesday when she wakes up. It's 1 of 2 situations for her. I think the paradox is created because, using the illustrations in the video, the fourth situation in the bottom right square where heads was flipped, it's Tuesday but she is not awoken is not talked about but it is still an event that has a probability of happening. But it seemingly doesn't "exist" or happen for her because she is not conscious.
This feels basically like nonsense. If both answers are "valid" and "accepted" (tbh kind of makes me cringe to hear a mathematician say that!) then the question is not well formed or the wrong tools are being applied.
The question has multiple interpretations and therefore multiple valid and accepted answers. It's a type of paradox, rather than some kind of mathematical statement, no nonsense going on here
There is a bayhsian updating here. There are not 3 situations, there are 4 with equal probability. You're missing the scenario where it is Tuesday and she isn't woken up. When she is woken up, she knows that possibility is eliminated, and so her probabilities update, knocking it from 4 equal probabilities down to 3.
This is one of the few numberphiles I've had to re-watch for comprehension. I LOVE the brain-strain here. I've got a variation for you to play Sleeping Beauty: "Sleeping Beauty is going to be put to sleep on Sunday. The experiment hosts will flip a coin after she goes to sleep and if it comes up heads, she will be woken up on Monday and asked a question and be put back to sleep with the memory of Monday wiped from her mind. if the coin comes up tails, she will be woken up on Monday and follow the procedure as if it were heads, however she will be woken up again on Tuesday and repeat the Monday procedure again, wiping the Tuesday from her mind. (all the same as the original experiment) She'll be woken up on Wednesday to end the experiment. However in this variation, she is given a coin which she is allowed to flip to help her acheive a 50% randomness to aid her answer, but she's only allowed to flip it once and must choose to either flip it before going to sleep on Sunday, or she will flip it upon being woken up on Monday as well as on Tuesday. She will not know on Tuesday that she already flipped the coin and will be able to choose to flip the coin on Tuesday as if she hadn't woken up on Monday. The question she will be asked each day she's woken up "Did the coin flipped on Sunday come up Heads or Tails." And she will be incentiviced to have answered correctly the most amount of times. This begs the strategy... Should sleeping beauty flip the coin before going to bed and always answer according to that result, or should she flip the coin after waking up and answer according to that result." It's a weird variation that I think plays at the question of stuborness vs flexability in the face of many unknowns.
This is deeper than I thought, I've flipped twice but now I'm confident it is 1/3 because if you keep score of his answers then that's the answer that will tally correctly over the long run.
PS. I looked at wikipedia where the question is "What is your credence..." which I think is subtly different to "What is the probability..." . It clarifies that things are to be looked at from the sleepers perspective.
Seems to me this is simply an issue with the ambiguity of language. Sleeping Beauty is not asked, "When this fair coin is flipped, what is the probability it would be tails?" But that's what we usually think is meant. Instead, she is asked, "Given _this experimental setup,_ what is the probability that we flipped this fair coin and got tails?" The two questions seem like they should be the same, but they aren't. The first question entails its own answer, as it is a fair coin. The second question is about the experimental design. In the Rip van Winkle extension, if Sleeping Beauty were to repeat this process (let us assume it is hours instead of days, simply to spare her the time!), then she will sleep for just about six weeks (41.666... days). We can then consider the experiment repeated for a year (about 8 times). On average, we would expect 4 heads and 4 tails--which means about 4 awakenings after 1 hour and then sleeping for 999 hours, and 4000 awakenings spread across every hour of the ~6 weeks. Even if she got a string of heads, such that the coin only came up tails a single time, the tails awakenings would still vastly outweigh the heads awakenings, e.g. 7 heads-awakenings vs 1000 tails-awakenings, despite that event being quite unlikely (3.125%). Hence, she should consistently bet on getting a tails-awakening, rather than a heads-awakening. Perhaps that's the best way to look at this question. It _sounds_ like we're being asked what the probability of the coin is, but we aren't. We are being asked what the probability of getting a heads-awakening is vs the probability of getting a tails-awakening. Consider what I will call the "memory wipe" variant. Sleeping Beauty wakes up, unsure of where she is. She is then told that she has just finished the experiment; it is now Wednesday and she is free to go. But, before she leaves, she is asked what her credence is for whether the coin was heads or tails. The only information she has gained is that she is done with the experiment--this awakening _is not_ contingent on the coin. Quite obviously, her credence must be 1/2, as she has not yet learned which path she took. _This_ question has returned to being about the coin itself--because, to use Brady's terms, she is no longer walking the path. While she is still _on_ the path, the question is about the path, not the coin.
Did you just ask chatgpt to hallucinate a solution for you? You added some fluff to that explanation that makes no sense! In all that text and baffling conversations of hours into years for no reason, you didn't explain why you would read the question the way you did. How can you see "how many awakenings happen on a tail flip" from the question as asked "what is the probability the coin was a head?" None of the permutations of the question in this video come close to what you seem to be assuming. But this is what makes me wonder if there is some version of the paradox out there that actually does ask a sufficiently ambiguous question to seem paradoxical, and this version just messed up the question a bit.
Here's my opinion as a mathematician: the situation is analogous to the difference between asking someone about the probability of the coin landing heads without showing them the outcome, and the same question after showing them that it landed tails, i.e. the difference between unconditional & conditional probabilities. The conditional information in the sleeping beauty problem (i.e. the fact that she has just woken up) is less strong than knowing the actual outcome of the coin flip, but still enough to make the probability quite different from the unconditional probability (i.e. 1/3 instead of 1/2). If we allow sleeping beauty to wait a couple of days before answering, then if she isn't put to sleep again she should answer 1/2. If sleeping beauty is told (after waking up) that she can win some money if she correctly guesses the coin flip then she is more likely to win if she says tails than heads, i.e. if we repeat the experiment multiple times with 2 different sleeping beauties; one of whom guesses tails and one who guesses heads, then the one who guesses tails will win more money. All those arguments about saying that the probability of the coin landing heads is always 1/2 regardless of her situation are disregarding the extra information that she has from knowing that she has just woken up, and so it's a bit like flipping a coin, seeing the outcome is tails, and then saying that the probability that a head was flipped is 1/2.
The correct question is: what is the conditional probability of heads given that SB is woken up. Let us calculate this: 1. Since the coin is fair, P(heads) = 1/2. 2..SB is always woken up, whether heads or tails. So P(SB woken up) = 1. 3. Since SB is always woken up after heads, the event "heads AND SB woken up" is the same as "heads", and P(heads AND SB woken up) = P(heads). Thus: P(heads | SB woken up) = P(heads AND SB woken up) / P(SB woken up) = P(heads) / 1 = 1/2. So 1/2 is the correct answer, being the conditional probability of heads given SB is woken up.
@@viliml2763 you haven't accounted for the day (which is random from SB's perspective)... in fact the probability of being woken up on a particular morning during the 2 day experiment is different depending on the coin flip; P(woken up|heads) = 1/2, P(woken up| tails) = 1. To be more explicit; P(woken up | heads) = P(woken up & 1st day | heads) + P(woken up & 2nd day | heads) = P(1st day | heads) + 0 = 1/2 + 0 = 1/2 P(woken up | tails) = P(woken up & 1st day | tails) + P(woken up & 2nd day | tails) = P(1st day | tails) + P(2nd day | tails) = 1/2 + 1/2 = 1
@@viliml2763 Well, if you're asking if waking up makes a fair coin into a not fair coin: obviously it doesn't. But the thing is that here (and in the video, since it seems to be ambiguous) P(heads) is not well defined. This is where the "paradox" arises. If P(heads) means "the probability that a fair coin comes up heads" you are absolutely correct and the answer is 1/2, regardless how often SB wakes up. But P(heads) can also be interpreted as "the probability SB sees at the moment for the coin having come up heads on Sunday", which would mean that you can't automatically assume 1/2 for P(heads). That in turn leads by lack of information to the need for another assumption, and the most popular one is "since we cannot distinguish the instances of waking up, let's assume they're equally likely", which like it's shown in the video leads to 1/3 as the answer. So as long as the question is not making clear what exactly it is we want to know, both answers can be argued to be correct.
@@viliml2763 Except that the conditional information isn't "SB was woken up at least once," it's "SB was woken up right now." "Right now" might be Monday or Tuesday, which changes the conditional probability of "SB woken up." Since time moves forward independent of our coin toss, P(heads AND Monday) = P(heads AND Tuesday) = P(tails AND Monday) = P(tails AND Tuesday) = 1/4. P(SB woken up) = P(SB woken up | Monday) * P(Monday) + P(SB woken up | Tuesday) * P(Tuesday) = 1 * 1/2 + 1/2 * 1/2 = 3/4 Our question is, as you said, P(heads | SB woken up) = P(heads AND SB woken up) / P(SB woken up) = P(heads AND Monday) / (3/4) = (1/4) / (3/4) = 1/3
When the coin is flipped, there's a 1/2 chance that the coin lands on heads and a 1/2 chance that it lands on tails. Therefore, if the experiment is repeated multiple times, then the number of experiments in which the coin landed on heads is equal on average to the number of experiments in which it landed on tails. The simplest version of this is that the experiment is done two times, with the coin landing on heads one time and the coin landing on tails the other time. Over the course of both experiments, she'll be woken up a total of 3 times. Two of the three times she's woken up, the coin would have landed on tails, while one of the three times she's woken up, the coin landed on heads. If she's asked what the coin landed on each time she's woken up, she should say 'tails' two of the three times and 'heads' the other time if she wants to get all 3 right. Therefore, the probability that the coin landed on tails on any given time that she's woken up is 2/3. Pretty easy, but a cool puzzle! I liked this one.
A lot of people reverse the conditions for her being woken Monday and Tuesday, I'm going with she is woken on Tuesday if it comes up tails and not woken on Tuesday if it comes up heads. Here is how I would motivate the paradox. So usually when you bet you assign your bets in preference to your sense of the odds (probability) therefore a bet should reflect the probability estimate. So if as Brady suggested (and occurred to me as I was watching) we include a payout related to her answer to the question "What do you think the coin came up?" asked when you wake her up on Monday and Tuesday with her getting $1 if she correctly guesses and nothing if she is wrong. In such a scenario the expected payouts for different strategies are easy to specify. If Sleeping Beauty absolutely favours heads then we expect her to get 50 cents (half the time $1 and half the time nothing). If she is indifferent to what she bets and before answering flips her own coin then we expect her to win 75 cents (this is a bit tricky to show but I think I got the combinatorics right). Finally if she absolutely favours tails we expect her to get $1 (half the time she gets $2 and half the time nothing). We can even turn it into a non-monetary bet, let's say Sleeping Beauty finds it very fun and exhilarating to be shown video footage of her being correct in a guess. The experimenters video tape all her Q&A sessions. She will be shown them on Wednesday. So on Sunday she knows if she commits to betting Tails she will either see one piece of footage where she wrongly guesses tails when it was head or two pieces of footage when it was tails (and she got it correct), so she commits to tails since it is the way to get the best chance of the most video of her winning. We might say she is maximizing her chances of being right by preferring tails (as if tails was somehow more probable), but it probably needs to be more precisely said she is maximizing her emotional elation at being right in the same sort of true stimulus/recording being played. Given that her winning strategy is guessing tails does that mean she has estimated the probability of tails as absolutely greater (and given the expected payouts she favours tails 2 to 1). If we really only laid our bets according to our understanding of the odds than yes. However my initial statement missed something betting reflects the odds all other things being equal. If you have different payouts for different bets that changes ones betting strategy or different numbers of winning opportunities. This is the entire reason that odds determine payouts, the booky gets people to bet on the long odds by having proportionally large payouts. If you can buy one of two kinds of tickets and type A has twice the payout of type B if it wins but the probability of A wining is the same as B you favour A. If you can buy twice as many type A tickets as B but the odds and payout per ticket is the same whichever you do, you buy twice as many chances to win (twice as many As) and so on. Regarding my idea of showing Sleeping Beauty video of her answers to elicit an emotional response. What if she is bored by having to see the same right answer twice, but it actually lessons her disappointment by acclimitization to see herself get the same wrong answer twice. Then she should favour answering heads. As it will be more agreeable to watcher herself get heads right once if she guesses that and it turns out right than the tedium of watching her get tails twice if its tails and she guesses that. Conversely it will be easier to take getting tails wrong by saying heads and seeing herself do the same wrong thing twice where the familiarity softens the blow to the sharp shock of guessing tails and seeing herself get it wrong only once. Edit: Likewise you can imagine she has emotional responses such that she enjoys the surprise of giving different answers if it lands Tails (one correct, one incorrect), so flipping a coin would be favoured if she finds that more enjoyable than two correct answers and for heads finds the 50% chance of correct balanced by 50% risk of incorrect. So depending on what objective facts you are tracking and what measure of correct you have lots of different answers.
For the "thirder" argument it is asserted that: P(T|Mon) = P(H|Mon) but if you work out the relationship assuming a fair coin you get that P(T|Mon) = 1/2 P(H|Mon) which would mean that: P(T∩Mon) = 1/4 P(T∩Tue) = 1/4 P(H∩Mon) = 1/2 P(H∩Tue) = 0 therefore: P(H) = P(T) = 1/2 See full reasoning below: The coin is fair: P(T) = P(H) P(T) = P(T∩Mon) + P(T∩Tue) P(H) = P(H∩Mon) + P(H∩Tue) Since (H∩Tue) does not happen: P(H∩Tue) = 0 Since you can't tell which day it is if it is tails: P(T∩Mon) = P(T∩Tue) Therefore: P(T∩Mon) + P(T∩Tue) = P(H∩Mon) + P(H∩Tue) P(T∩Mon) + P(T∩Mon) = P(H∩Mon) + 0 2P(T∩Mon) = P(H∩Mon) Since P(A∩B) = P(A|B)P(B): 2P(T|Mon)P(Mon) = P(H|Mon)P(Mon) P(T|Mon) = 1/2 P(H|Mon)
I believe the discrepancy here comes from a lack of clarity: in that section P(T) and P(H) should not imply P("a fair coin toss comes up heads/tails") respectively, but P("I got woken up just now because the coin toss came up heads/tails on Sunday"). - which is absolutely not clear from the notation and I only got that idea from what I understood about the context. But with that you can't automatically assume P(H) = P(T) = 1/2 and that with no other information one would guess P(T|Mon) = P(H|Mon) seems reasonable to me.
It’s 50/50. The amount of times you wake her up doesn’t affect the probability. I understand why people think this is a paradox, but it just isn’t. Yes, when you wake beauty up it could be any of the three possible wake up times (two on Monday, one on Tuesday), but two of those wake up times share the same 50% chance of the initial equation.
I think the problem though is that Monday tails and Tuesday tails are not separate events because they are a result of the same event (one coin flip) that just spans two days long (two wake ups instead of one) and I think it's a fallacy to refer to Monday tails and Tuesday tails as separate events.
Tom could be just doodling nonsense on the paper and he would come up with a way of making it interesting. He just has so much passion for what he is doing :)
12:00 There is no difference between the coin flip vs. "the pathway that resulted from the coin flip". If the coin is heads then you end up on the heads path, and you only end up on the heads path if the coin is heads. The flip result and the path are equivalent.
@@5h5hz Except that the claim that the question can be interpreted in two different ways is incorrect. Only 1/3 is correct the way the question was phrased.
@@5h5hz Run the experiment 100 times, 50 heads, 50 tails. Sleeping Beauty will be woken 150 times. 50 times Monday following heads, 50 times Monday following tails, 50 times Tuesday following tails. P (Heads) = 50/150 = 1/3.
I think the best way to explain the problem here is the fact that what sleeping beauty is trying to do while answering is not fully defined. Its basicly a question of how sleeping beauty measures her own success. Aka what her goal is. If her goal is to be right the maximum amount of times she will answer one way. If her goal is to be right at least once, she will answer with 50/50. As with a vast majority of "paradoxes" it is simply the fact that the problem isnt defined enough. Resulting in multiple answers depending on how you fill in the missing information.
Of course the video describes changing what is being asked to fully constrain the problem. But I think changing sleeping beauty's goals is a much more obvious proof that the problem isnt defined enough. Since the wording of the question doesnt need to be changed in order to explain it.
The "Sleeping Beauty Paradox" is such a mind-bending concept, and this Numberphile video does a fantastic job of explaining it in a clear and engaging way. It's one of those philosophical puzzles that can really make you question your intuitions about probability and decision-making. The discussion and different viewpoints presented here add an extra layer of intrigue to the paradox. Numberphile consistently delivers thought-provoking content, and this video is no exception. It's a great reminder of how math and philosophy can intersect in the most perplexing and fascinating ways.
I think the paradox is in the question. The question is asked 4 times, two on Mon and Tue (in Tail case), and two on Mon and Tue (while asleep) (in Heads case). So if each has 1/4 a chance, head/tails is 50/50, at thesame time 1/4 Head ((while awake)) is 1/3 of the 3/4 ((of questions while awake)).
Personally I thought it's 1/3 at first but then I gradually convinced myself it's 1/2. Because it's like knowing beforehand that she would wake up in one of 2 universes - the one in which the heads dropped or the one in which the tails dropped and the number of wake-ups doesn't matter
I agree. Thirders are full of themselves if they think that a perfect coin toss changes its probability based on the number of times you check the result.
That's a matter of number of wake-ups, not the probability itself. If I would pay you 100 times as much if your roll 2 sixes on dice than I would for any other result, it doesn't change the probability of double six being 1/36. Neither would paying it x times.
Halfers answer as a third-party perspective of the coin flip. Thirders answer from Sleeping Beauty's perspective where the coin flip doesn't exist half the time for a heads result.
As multiple peope here and in video already said: its not really a paradox, its just a an interpretable question. The good thing about probablistic questions: we can just simulate them. And when you ask the sleeping brady what the probability for Tails is on the flip on saturday, the answer is 50%. When you ask predict the flip on saturday the sleeping brady is correct 2/3 of the times with the answer "Tails". And thats just because you ask him more often in the "Tails" case. I really dont get why this should count as a paradox. There is nothing "paradoxic" about it, both cases are pretty clear cut. Ambiguous questions dont gerenate paradoxe. And its not even called a "paradox" on wikipedia.
I feel like those assumptions under the thirder argument are what mess it all up. I prefer Brady's explanation: it's a 50/50 chance that the coin was a heads, but it's a 1 in 3 chance that she's in that specific situation.
Why do you think the assumptions "mess it up"? Consider the two possible interpretations of the ambiguous question: Asking "is this fair coin fair"... is a ridiculous question with a trivial answer. Asking "how would you bet on the outcome" ... makes use of all the effort gone to running the experiment.
So great to have Tom Rocks Math[s] Tom back in an episode of Numberphile! This one feels like a cross between the Monty Hall Problem (referenced briefly in this video) and the Prisoners' Dilemma
In the original formulation of the sleeping beauty problem, The Monty Hall Problem is of relevance. The Problem was invented by Arnold Zuboff and named by Adam Elga. Please read His Important work.
I love this problem. It's both intuitive and easily provable that the probability that any coin on any day must be equal probability to any other coin and day, but the answer to the question depends entirely on whether Sleeping Beauty understands the whole experiment when you ask and what specifically you ask her.
Except that P(H on Mon) = 1/2, P(T on Mon) = 1/4, P(T on Tue) = 1/4. That in the tails case you are asked the question twice does not mean it is more likely that the coin came up with tails. In the tails case, when you wake up to Mon or Tue it does not matter that you are also woken up on Tue or Mon. You can basically ignore the other waking up. If it happened yesterday, you don't remember it anyway. If it happens tomorrow, you won't remember today anyway.
This is very interesting and I think it is very simply resolved by specifiying the question in more detail. 1. What is the likeliness the coin flipped heads? : It is 1/2 2. Considering you just woke up, what is the likeliness the coin flipped heads?: 1/3 It is way more fun and paradoxical if you just ask the 1st question.
Brady, you nailed it brother. The correct approach depends on what question sleeping beauty is asked. 1. Was the coin flip heads or tails? Halfers are correct. 2. What day of the week is it? Thirders are correct.
Cool. The action of waking up is transforming information. Consider sleep as 0 and wake as 1. This is what changes the outcome. So, if we write a truth table and consider the time she was asleep and when she woke, the probability is 1/2. Therefore, saying that she has no information is false because, as observers, we know that she both slept and woke up, which provides us with information.
Imagine being in a scenario where you're flipping coins in a room. By using the status of the coin to turn on/off the light(0,1), you observe 1 head (H) and 2 tails (T). However, you're deliberately excluding the times when it's heads half of the time. This intriguingly means there is no randomness involved. Here's the kicker: you have the ability to force the outcome to be any desired proportion. It's fully controlled . Consequently, you end up observing heads (H) one-third of the time and tails (T) two-thirds of the time. Nevertheless, this doesn't alter the probability of the coin itself, which remains 1/2 throughout.
The probability is 50:50 since P(T|Mon) != P(H|Mon). There are no reason for these two probability should be equal and P(T|Tue)=1!= P(H|Tue)=0. Interesting mistake that people would make is that asking multiple times duplicate the number of cases. But actually P(T&Mon) = 1/4 and P(H&Mon) =1/2
To me this feels a lot like that WWTBAM meme question, What is the chance you get the correct answer if you randomly choose an answer? A 50%, B 25%, C 25% or D 75%.
The issue here is that there are two different meanings for the word probability. Before the experiment the coin has a 50/50 chance of landing on either side, and there is no real fact of the matter. But when sleeping beauty is woken up the flip has already happened, so it must have landed on one side or the other and it doesn't really make sense to talk about the probability. Instead the question is asking about sleeping beauty's best guess of what happened given her knowledge of the situation, which gives the 1/3 answer. Of course you could insist that the question is really about the initial 1/2 probability, but then there would be no point in the whole sleeping beauty story since whatever you plan to do arter the flip obviously has no impact on the probability before the flip. So its pretty clear that isn't what the question is going for, especially with the original "credence" phrasing.
Ok, now, imagine the sleeping beauty's friend experiment: you perform the sleeping beauty experiment to conclusion as usual and then ask her friend the probability she was woken up Tuesday during the experiment. As Brady points, asking her multiple times is what causes the conflicting answers.
Asking multiple times only matters if you ask about the expected number of correct answers. We're asking about the probability so it's equivalent to only asking once.
What a great video! I might be totally wrong, but it brings to my mind things like the anthropic principle, doomsday argument and Boltzmann brain problem. While I know the objective probability of heads and tails is 50/50, I can't help but find myself a thirder here!
@@maxonmendel5757 Anthropic principle : if I'm alive, then the universe has to be just right for my existence to be possible (which doesn't mean it was fine tuned). Doomsday argument : if I'm alive, assuming I'm one of the humans in the "most populated century" since that's the most probable century to be born in, then humanity should go extinct soon (but one has to remember that this is based on an assumption so can hopefully be wrong) Boltzmann Brain : if I'm alive, it's very probable that I'm just a brain that popped into existence by random fluctuations and is generating random inputs that happen to make me experience a fake reality (but although it's more probable, reality seems fun so let's go with it)
@@ghislainbugnicourt3709 Huh, the version of the doomsday paradox I’m more familiar with is, “there a (e.g.) 90% probability that I’m in the middle 90% of all-humans-who-will-ever-be-born (ordered by date of birth), and therefore the number of humans who will be born after me is less than 95% of them, and so there will be at most 20 times as many humans born in the future as were born before me” But, same general idea...
I totally agree... the objective, external reality is that there is an equal probability of heads and tails, but the subjective reality is that it's more likely you're in a scenario where the coin toss result was tails.
And I think the two questions are: "What are the odds that the coin came up heads?" vs "what are the odds that this is the first time you've been woken up?" (first can be replaced by any number, but 1st is 0.2% and every other number is 0.1%). That's the only question that actually has a thousand possible options to choose from.
Tom derived one equality from one assumption, then another equality from a different assumption, then because one side from each equality was the same statement he then combined them into a threeway equality, but surely you cannot do that if each of the two equalities required a different assumption, because now the threeway equality surely requires both assumptions to be true or else it isn't valid, and making both assumptions simultaneously would sort of collapse the problem into a certainty that it was Tails and a Monday wake up anyway.
He removed the assumption by going from the conditional probability to the probabilty that both events happen. Where the argument goes wrong is in P(T|Mon) = P(H|Mon). If you consider being woken up as the event, then if you know you woken up on Monday, it is more likely to be heads (2/3) than tails (1/3).
@@ronald3836 If you were woken up and told it was Monday (aka you know it is Monday) then surely the probaility is 50/50 for Heads or Tails, because the additional Tuesday wake up for the Tails path wouldn't come into it
@@MrDannyDetail No, there are 3 cases: H on Mon, T on Mon, T on Tue. The probability of H on Mon is 1/2. The probabilities of T on Mon and of T on Tue are both 1/4. This means that P(Monday) = 1/2+1/4 = 3/4, P(Tuesday) = 1/4. And P(T|Mon) = P(T and Mon)/P(Mon) = (1/4) / (3/4) = 1/3. P(H|Mon) = P(H and Mon)/P(Mon) = 2/3. What tricks the mind is that you are woken up both on Monday and on Tuesday in the tails case, but since you have amnesia there is no difference if you change that rule to being woken up either on Monday or on Tuesday with equal probability. As the video correctly stated, given that it is tails, the probability of it being Monday is equal to the probability of it being Tuesday, so both 1/4.
I think the best way to visualize this is with a tree. Imagine there is an apple tree with a distinct shape, its trunk splits in half, forming the shape o a “Y”. On the right trunk there are 10 apples growing and on the left trunk there are 20 apples. If you pick one of the apples, what is the probability of it coming from the right/left trunk? This mimics the 1:2 ratio of SBs problem (Right = Heads/Left=Tails) but this way it's clear why there are two answers, it depends on HOW you chose to pick the apple. If you look at the 30 apples and choose one of them, there is a ⅓ chance of it coming from the right, but if you first choose which trunk to climb (50%) and THEM pick an apple at random from that trunk, there is a ½ chance of it coming from the right. If you pick the trunk you’re going to climb without looking at how many apples there are in each, the chances of it coming from any given trunk are the same as the chances of picking that trunk to begin with, no matter how many apples there are in each trunk. On the other hand, if you look at all the apples and decide which to pick and then use that to deduce the truck you have to climb, then that's the same as picking all the apples, putting them in a basket, choosing one at random and asking whether it came from the right or left. Those are two FUNDAMENTALLY different things, but it's hard to realize that in SBs version of the problem.
This is a cleverly devised Monty hall problem. At 4:24 it is correctly stated that she "doesn't learn any new information by being woken up" yet at 0:14 we are casually informed that she "knows all the details of the experiment". This is all the information she needs to be a thirder like me. Furthermore, Tom goes on to explain that it is a language ambiguity which as it happens is often the cause of paradoxes, and would be correct if she hadn't been given the additional information. It all goes to show just how carefully questions and suppositions need to be analysed and how easily influenced we can be by presentation. I love this video.
The Monty Hall problem starts with three equally likely probabilities, and then two of them are collapsed into one, leaving two unbalanced choices. This is the opposite: it starts with two equally likely probabilities, then one of those two is split into two possibilities, which thirders erroneously set as each having the same likelihood as the other option.
The assumption made @7:00 in the video confuses me and I’m not sure it’s valid. Instead of making a different assumption about those conditional probabilities, I’m going to go through all the long math in this comment to explicitly calculate for the probabilities Tom assumes here are equal. With this, I will attempt to show how these two seemingly opposing views can be made more consistent. Setup and Events: Mon=It is Monday Tue=It is Tuesday H=Heads T=Tails Note that H and T are Complements and so are Mon and Tue. Assuming fair coin: P(H)=1/2 P(T)=1/2 Calculations: P(Mon|T)=P(Mon AND T)/P(T) =>1/2=P(Mon AND T)/(1/2)=2*P(Mon AND T) ==>P(Mon AND T)=1/4 P(Tue|T)=P(Tue AND T)/P(T) =>1/2=P(Tue AND T)/(1/2)=2*P(Tue AND T) ==>P(Tue AND T)=1/4 ==>P(Mon AND T)=P(Tue AND T)=1/4 P(Mon|H)=P(Mon AND H)/P(H) =>1=P(Mon AND H)/(1/2)=2*P(Mon AND H) ==>P(Mon AND H)=1/2 P(Tue|H)=P(Tue AND H)/P(H) =>0=P(Tue AND H)/(1/2)=2*P(Tue AND H) ==>P(Tue AND H)=0, this one is trivial P(H|Mon)=P(Mon AND H)/P(Mon) =>P(H|Mon)=(1/2)/P(Mon) (*) P(T|Mon)=P(Mon AND T)/P(Mon) =>P(T|Mon)=(1/4)/P(Mon) By (*), P(T|Mon)=1-P(H|Mon)=1-(1/2)/P(Mon)=(P(Mon)-(1/2))/P(Mon) =>P(T|Mon)=(P(Mon)-(1/2))/P(Mon) => (1/4)/P(Mon)= (P(Mon)-(1/2))/P(Mon) =>1/4=P(Mon)-1/2 ==>P(Mon)=1/4+1/2=3/4 ==>P(Tue)=1-P(Mon)=1-3/4=1/4 ==>P(Mon)=3/4 and P(Tue)=1/4 (*)==>P(H|Mon)=(1/2)/(3/4)=2/3 ==>P(T|Mon)=1-P(H|Mon)=1-2/3=1/3 ==>P(H|Mon)=2/3 and P(T|Mon)=1/3 P(H|Tue)=P(Tue AND H)/P(Tue) =>0=0/(1/4)=0 trivially P(T|Tue)=P(Tue AND T)/P(Tue) 1=P(Tue AND T)/P(Tue)=(1/4)/(1/4)=1, so all conditional probabilities in this setup are consistent. Results: P(Mon)=3/4 P(Tue)=1/4 P(Mon AND T)=P(Tue And T)=1/4 P(Mon AND H)=1/2 P(Tue AND H)=0 P(H|Mon)=2/3 P(T|Mon)=1/3 P(H|Tue)=0 P(T|Tue)=1 Therefore, we see that the assumption made in the video that P(T|Mon)=P(H|Mon) is false. While intuitively it’s true, we have calculated the probabilities to be 1/3 and 2/3 respectively, and thus knowing it’s Monday does clearly update your probability on what side of the coin was flipped. This is more like the “halfer” view in the sense that assuming nothing of the day, we still have 50% for the coin flip. However, we do get probabilities of 1/3 and 2/3 now, except here they represent the probability for Tails and Heads GIVEN you know it’s Monday so not completely the same as the “thirder” view. It’s still paradoxical to me though that knowing it’s Monday should affect the probability when both situations would have sleeping beauty inevitably being woken on Monday anyway. But this is just our intuition failing us, and the math tells us that Monday pushes the probability more in Heads’ favor. Please let me know if I made a mistake in any of my calculations.
Setup: A=Now it is Day 1 B_i=Now it is day i for i in {2,3,4,…,N} The B_i’s behave identically so we will use B to refer to any of them. H=Heads T=Tails H*=T A*=UB Where * here is complements and U is union. P(H)=1/2 P(T)=1/2 N=amount of times woken up if Tails is flipped Calculations: P(A|T)=P(A,T)/P(T) 1/N=P(A,T)/(1/2)=2*P(A,T) ==>P(A,T)=1/2N P(B|T)=P(B,T)/P(T) 1/N=P(B,T)/(1/2)=2*P(B,T) ==>P(B,T)=1/2N ==>P(A,T)=P(B,T)=1/2N P(A|H)=P(A,H)/P(H) 1=P(A,H)/(1/2)=2*P(A,H) ==>P(A,H)=1/2 P(B|H)=P(B,H)/P(H) 0=P(B,H)/(1/2)=2*P(B,H) ==>P(B,H)=0 P(H|A)=P(A,H)/P(A) P(H|A)=(1/2)/P(A) (1) P(T|A)=P(A,T)/P(A) P(T|A)=(1/2N)/P(A) P(T|A)=1-P(T*|A)=1-P(H|A)=1-(1/2)/P(A)=(P(A)-(1/2))/P(A) ==>P(A)>1/2 ==> (1/2N)/P(A)= (P(A)-(1/2))/P(A) ==>1/2N=P(A)-1/2 ==>P(A)=1/2N+1/2=1/2N+N/2N=(N+1)/2N ==>(N-1)P(B)=1-P(A)=1-(N+1)/2N=2N/2N-(N+1)/2N=(N-1)/2N ==>P(B)=1/2N (1)==>P(H|A)=(1/2)/((N+1)/2N)=N/(N+1) ==>P(T|A)=1-P(H|A)=1-N/(N+1)=(N+1)/(N +1)-N/(N+1)=1/(N+1) P(H|B)=P(B,H)/P(B) 0=0/P(B) ==>0=0 P(T|B)=P(B,T)/P(B) 1=P(B,T)/P(B)=(1/2N)/(1/2N)=1 Results: P(A)=(N+1)/2N P(B)=1/2N P(A,T)=P(B,T)=1/2N P(A,H)=1/2 P(B,H)=0 P(H|A)=N/(N+1) P(T|A)=1/(N+1) P(H|B)=0 P(T|B)=1 Case N=2: P(A)=3/4 P(B)=1/4 P(A,T)=P(B,T)=1/4 P(A,H)=1/2 P(B,H)=0 P(H|A)=2/3* P(T|A)=1/3* P(H|B)=0 P(T|B)=1 Case N=1,000,000: P(A)=1,000,001/2,000,000~1/2 P(B)=1/2,000,000~0 P(A,T)=P(B,T)=1/2,000,000~0 P(A,H)=1/2 P(B,H)=0 P(H|A)=1,000,000/1,00,001~1 P(T|A)=1/1,000,001~0 P(H|B)=0 P(T|B)=1 Case N=2(Original): P(A)=3/4 P(B)=1/4 P(A,T)=P(B,T)=1/4 P(A,H)=1/2 P(B,H)=0 P(H|A)=2/3 P(T|A)=1/3 P(H|B)=0 P(T|B)=1
@@RaRa-eu9mw I defined the events and did the probability calculations. I didn’t assume anything of the situation under any “point of view.” How are these calculations then not consistent under the “thirder”point of view? Where does the math go wrong?
@@Happy_Abe Sure. The issue is that the initial P(H)=1/2 is the probability that a fair coin lands on heads, but it then appears in the calculations as the probability that, when Sleeping Beauty is being asked the question, the coin landed on heads. For the thirder point of view, P(H)=1/3 and so we get statements like P(Mon AND T)=1/3.
@@RaRa-eu9mw I’m not following you, we know it’s a fair coin so we start with the fact that P(H)=1/2 I’m not assuming this, we are given this information by the case of a fair coin. There’s no calculation I see(that doesn’t rely on further assumptions like made in the video that I addressed in the beginning of my comment) that would imply that P(H)=1/3. The probability that is actually 1/3 is the probability that Tails was flipped GIVEN it is now a Monday and there’s a 2/3 probability that Heads was flipped GIVEN it is now a Monday. Please provide the mathematical calculations showing that P(H)=1/3 without assuming that P(H|Mon) and P(T|Mon) are the same like in the video with no justification other than being intuitively true. While intuitively it makes sense to assume that, the calculations I did rely on no assumption and imply the contrary and are therefore more valid because they were derived, not assumed on intuition.
Yep, those two things are answering fundamentally different questions. Which precludes both solutions being correct. Only one can be correct per the question that is asked
Isn't there a destinct difference between "What is the probability the coin was head?" and "Do you believe you are on a 'head'-branch of being woken up?"? I mean the difference is "what is the probability the coin was head?" vs "what is the probability the coin was head given we wake you up?" The second question is close to what Brady said with the bet of getting x value every time you are right, meaning the expected value is higher saying you are on the branch with more wakeups. Also I don't think it can be healthy for sleeping beauty to get this medicine 999 times 😅😂
One of the coolest things I ever saw was a math teacher prove that 1=2. He had to divide by zero to get there, which is not possible in math. I think thirders are "dividing by zero". A coin toss is always 50/50. In the same way, the effect can never "effect" the cause of the effect.
If you do get extra information, then the conditional probability of heads given that information could be different from 1/2. But in this case the point is that you do not get extra information, so I completely agree with 1/2.
I toss two coins and hide them. I tell you that the coins under my hands are not two heads. What is the probability that the first coin is a head? I have a bomb wired to a fair 6 sided dice, so that if it rolls a 1, it will detonate, killing both of us. I roll the die and ask you what the roll was - what is the probability it was a 6 (same question for each other number)? What is the probability it was at least a 2? By eliminating a result, *and knowing that result is eliminated*, you have changed the game and the conditional probabilities *based on the information you have*.
This just seems like an incomplete question, and Brady started to touch on that towards the end. The example he gave is "every correct answer gets you $1000" to which the answer would of course be tails, but the counter part is "if you were correct every time, then on Wednesday you get $1000" then your answer wouldn't matter as long as it was consistent (which the possibility of a changing answer is a separate rabbit hole). It reminds me of the video with Grant about choosing a "random" chord on a circle.
Is it just me or does the second assumption (coin flip given monday 50/50) just seem false? You can get the values without assuming this: The distribution of the initial coin flip should be given at the start. If it isn't stated explicitly I assume it's 50/50, that is P(T)=P(H)=1/2. I understand the first assumption. Conditioned through the coin being Tails, we're looking at a particular experiment where you definitely woke up twice. So in that experiment there's no reason to believe the current day is more or less likely to be a monday. This would give is P(Mon|T)=P(Tue|T)=1/2. Okay, we don't need to assume anything else: The probability P(Mon and T) is simply P(T)*P(Mon|T)=1/4. Same with P(Tue and T). However, P(Mon and H)=P(H)*P(Mon|H)=1/2*1=1/2. That means without assuming anything else we get back probability of heads = P(H)=1/2 (as we should). Assuming P(H|Mon)=P(T|Mon) is creating a contradiction with their real values: P(H|Mon)=P(H and Mon)/(P(H and Mon)+P(T and Mon))=2/3, and P(T|Mon)=1/3. To me there's no paradox, the other value arises from an assumption that results in a contradiction.
I feel there's some sort of unsettling implication to this problem: Are we usually the Sleeping Beauty, trying to find the truthful probability of an event from a biased position? We could do perfect math, but still the answer could be wrong because we can only look at things from our perspective.
That's a great Insight. The sleeping beauty problem was actually invented by Arnold Zuboff and named by Adam Elga. In Zuboff's Work, he talks about the perspectival Nature of probability. This is important to the solution of the sleeping beauty problem.
You are totally right. The universe doesn't pick if you win on a lottery ticket when you scratch it. It is determined when it is printed, delivered and stored. By the time you purchase the one on top, it's no longer probability unless YOU ignore the content of the pile. You don't know the outcome yet, but that outcome now has a 100% chance of occuring, assuming you scratch that ticket at some point in the future.
I reckon the thirder aproach is the correct one. Yes, in a vaccum, the coin is fair, so 50%. But, with the rules of the game, the sheer fact that she is woken up is information itself. Imagine, instead of being woken 1 or 2 times, we make it so if the coin was heads, she is woken once; if tails, not at all. If she is then woken up, she knows for a fact the coin was heads, or she wouldn't be awake on the first place. The original situation is subtler, but the fact that she is woken twice on a head makes it so there are more opportunities for her answer heads. 2 out of 3 times she is awoken, the coin is heads, since she can't know which of the three posibilities it is. It is the rules of the game which affect the probability, not the fairness of the coin.
I feel like this is kinda similar to the observer effect in quantum mechanics. The simple fact that the question is asked to someone involved in the problem and thus "inside" of the probability tree itself, changes the meaning of the question and therefore its answer. Like if the probability itself was observer-dependent.
Rather than extend the experiement to 100 days, imagine instead that the experiment only lasts 1 day. So, if the coin flip is tails, sleeping beauty is woken up 1 time, but if the coin slip is heads, she is woken up 0 times. Then of course, if she is awake, she knows with complete certainty that the flip was tails!! But the "halfer" logic would says it's still only a 50% chance of tails! Point is, the fact of being awake does present new information, while the "halfer" argument depends on a total absense of new information. Going back to the original experiement, P(Heads)=P(Heads|Monday)*P(Monday)+P(Heads|Tuesday)*P(Tuesday) = 0.5*P(Monday)+0*P(Tuesday), so P(Heads)
I remember posting this one to Metafilter back in 2004 or so. The comments section eventually came to the conclusion that there is no paradox, it's just a badly defined question that we're asking Sleeping Beauty.
Completely agree. I had a rant about it in the comment section on Veritasium's video about it back in February. If it was voice acted, the voice would be laud and angry 🙂. Quoting myself: "Basically - the only problem I can see, is the problem that whoever came up with this *really* can't ask questions. The discussion about 50% vs 1/3 is more about *what the question is* than anything else. There's a reason I *kinda* prefer computer science over g.d. philosophy. " - Me, Feb 2023. I was... not happy. Currently the vote on Veritasium's video is 193,340 "Thirders", 85,660 "Halfers".
@@phizc I would expect all mathematicians sufficiently trained in probability theory to agree that the question asked is P(heads | you woke up), and that, since you always wake up, P(heads | wake-up) = P(heads AND wake-up) / P(wake-up) = P(heads) / 1 = 1/2. It is interesting that this simple exercise in conditional probabilities has become an accepted philosophical problem that has attracted dozens of publicatons, many of which are full of quite silly statements. This shows that it is best to ignore all the hand-wavy "probability" arguments by philosophers about whether we live in a simulation and similar questions.
It's actually very clearly defined, the question we are asking is P(heads | awake-now) - absolutely no philosopher or mathematician who has ever written on this problem disagrees.
@@dominiks5068 except that the philosophers are typically unable to calculate this simple conditional probability 😃. I agree with you that it is the correct formulation of what is asked.
Interesting problem, but I believe I agree with most people here that there isn't really a paradox. The correct probability is 1/2. The entire "thirder" argument is geared towards answering the different question "what is the probably that you are awake now AND the coin was heads." The betting argument is also good, where it's clear the thirder strategy is the better one but only because when the coin is T you get to "play" more often. Consider a modification: since Sleeping Beauty never gets any new information, we may assume that she will give the same answer every time. So since all the wake-ups under tails are dependent, they are all combined into one event (they all get the same answer and either all happen or none do). Now the 50-50 answer is pretty clear. So I think that's my final answer: the probably is 50-50 and the other strategies are all answering (or strategizing for) different questions. Cool problem, but not a true paradox in the sense of something self contradictory (I've never liked the other types also getting called "paradoxes" but that's not up to me).
It's not only about the question but about how her answer is evaluated. If you want to count how often she is right over all wake ups, she should say tails (correct chance is 2/3). If you count how often she was right in all wake ups after a toss she can say either (chance is 1/2). So the question is: Is one run off the experiment considered one wake up or one toss?
The simple answer that I came to with this is that you asking the question does not retroactively change the probability of the coin flip. Even in the 999 scenario, the probability of a coin flip is 50/50, no matter what. I'm with Brady on this one. Perception does not alter the statistics, and if no new information is being conveyed (which is the point of the amnesia) then there is no way to know whether it was a heads or a tails. It's 50/50 regardless. Funnily enough, this is like the opposite of the Monty Hall problem but with a similar solution. People are perceiving 2 doors and assuming the chance is 50/50, when in fact the probability was decided when there were 3 doors, so the odds are in thirds. This is the opposite. The perception is that there are 3 situations you could be in, but in reality there are only 2.
I feel like most of these mathematical "paradoxes" are the result of ambiguity in the question being asked. Even here, it was "well, if you ask it this way, it's 50/50 but if you ask it this other way then it's 1 in 3...or 1 in 1000".
This is similar to The Monty Hall Problem problem, which is that there are two questions, which can be expressed in very similar ways - subtle details distinguishing one from the other - and they can easily be confused. Once one question is asked, though, it has one answer. The listener might be expecting a different question, but if they correctly answer a question that's materially different from the one asked, they have not correctly answered the question asked. And the asker might have in mind one question, but stumble on that subtlety when putting it into words, and so the asked question can be different from an intended question. And it can even happen that both inversions happen, the intended question gets its correct answer via the asking of a different question. But once one question is asked, it has one answer.
If sleeping beauty is asked to make a wager which will be evaluated once on Wednesday (and if it was tails and she answered differently the different times she was asked, her most recent wager is the one which is used (alternatively, if tails, she is only asked on one of the two days)) then it makes sense for her to use 50/50 . So, if what wagers will be available and how they will be evaluated, is known beforehand, there are no paradoxes in “how should she bet”. The only difficulty is the question “what types of wager odds should we consider to define the probability?”. Here’s a variant: She is told beforehand that an amount of money X has been chosen, but she won’t be told the value of X until Wednesday. If heads, she will be offered a wager for an amount which is of size 2X, but she won’t be told the amount (as this would give information about whether heads or tails), and if tails, she will on each day be offered a separate wager of size X (but again, will not be told that it is of size X rather than 2X). (The wagers are each about which way the coin landed) In this case, the wager odds she should accept would be those in line with a 50/50 chance. From the half-ist perspective, the weightings on both sides are equal, so the way-to-bet is unchanged from the true 50/50 and from the third-ist perspective, the weighting added to the heads case shifts how one should bet away from the 1/3 probability to become 50/50 . Another variant: Instead of whether she wakes up twice, the coin only determines whether she will be offered on Tuesday the option to bet. (If heads, on Tuesday she will not be offered the wager.) In this case, it seems that there are 4 equally likely scenarios, {heads,tails}x{Monday,Tuesday} But, on the days she is given the option to bet, the information she has available based on which to make the bet, is identical to the information she has in the normal version of the scenario, so it sorta seems like the probabilities should be the same. In this case, P(Heads|monday and offered a bet )=P(tails|monday and offered a bet) But P(Heads|Tuesday and offered a bet) = 0, and P(tails|Tuesday and offered a bet) = 1 P(Monday) = P(Tuesday), but P(Monday|offered a bet) = (2/4)/(3/4)=(2/3) And P(tails|offered a bet) = (2/4)/(3/4)=(2/3) So, in a sense, if “am awake” in the original formulation is substituted for “am offered a bet” in this variant, it kind of seems like the third-ist position is essentially conditioning on “I am awake” as evidence about what day it is? Hm. I’m undecided as to whether this is “an actual question which has an answer”, or whether it is a question of semantics of “what do we want these words to mean” where the actual “how-things are” is clear.
There is no paradox whatsoever. The correct answer is 0.5, since there is only one independent event in the experiment - the coin toss. Therefore the probablility of waking up with heads and tail is always 1/2. The probability of waking up on Monday with Tails is 1/4 probability of waking up on Tuesday with Tails is 1/4 and probability of waking up on Monday with Heads is 1/2.
I'd say the halfer answers the question "What is the probability of a correct answer if we asked if the coin landed on heads (or tails)" While the thirder answers the question: "What is the expected number of correct answers if we asked if the coin landed on heads (or tails)". These are not the same question, hence different answers.
I was a thirder, but now I'm a halfer. But I think the way if interpreting it is as saying heads and getting it right gives you 1 valid answer and saying getting it wrong gives you 2 wrong answers. So it's a 50/50, but the expected value of right answers is 1/3.
I'm fascinated by the distinction in Brady's head, between "what is the probability that an event happened" and "what is the probability that we're on the branch where the event happened". To me they seem like the same thing. I feel like he's conflating the first one with "before the event happened, what _was_ the probability that the event was going to happen?". But that's different from asking "what _is_ the probability that it _did_ happen?" The literal answer to the second one is 100% (if it happened) or 0% (if it didn't). But what we usually mean is "what level of confidence should you assign to it, _given the information you have available_ ?"
Ah, the confusion between the question she was asked, "What is the probability the coin came up heads?" and the question she was *not* asked "What is the probability that a coin flip will come up heads?"
The first asks for the conditional probability given that you woke up. Since you wake up always, the answer is still 1/2. P(heads | wake up) = P(heads & wake up) / P(wake up) = P(heads) / 1 = 1/2.
@@reidflemingworldstoughestm1394 no, the conditional probability is calculated as I indicated. The outcome is 1/2. Translated into words: since you know beforehand that you will wake up, you don't gain any information from waking up. So the conditional probability of heads given that you woke up is equal to the probability of heads, which is 1/2 (fair coin).
@@reidflemingworldstoughestm1394 I like the way that Ronald explains his answer with reasoning and equations, and you just assert your answer with nothing but bluff and bluster. I guess the two of you are equally likely to be right! ;)
"What is a probability that the coin flip was heads?" is a very different question than "What is the probability that when you say the coin flip was heads, you are correct?" In the first case, the answer is 1/2 and in the second one, the answer is 1/3.
I'm still staying with the 50% 50% answer.The explanation using probabilities arithmetics just tells you that the 3 scenarios are equally likely to occur, but that doesn't mean that they are separate probabilities, so tthey shouldn't be treated like they can occur independently, it can't be Tuesday if it wasn't Monday first, they're both linked, so yeah, their odds are the same, because they're literally the same event, just at a different time.
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Dr. Zomboss SFX at 0:25 made me frantically look for a melon-pult
@@polyacov_yury2023 年 8 月 18 日. Have you 🇨🇳 read about the Reimann solution yet?
Brady hit the nail on the head, "What is the probability the coin was heads?" is a slightly different question from the question "What is the probability that sleeping beauty is woken and the coin was heads?", which is the question that you're always actually asking sleeping beauty, since she has to be awake to ask her.
I agree, there is no paradox. The answer depend on the question.
Sometimes all that is required is common sense.
The catch is that the question is NOT what YOU think about the probability that the coin was heads, but what sleeping beauty thinks about that probability when see is awake. That's why it's a paradox, got indeed a bit confusing as the interview went on.
i think the answer depends on the observer, (coin, person, etc), from the point of view of the coin is 1/2, from the pov of the person is 1\3
More precisely "What is the probability the coin was heads, given that Sleeping Beauty is awake?"
Asking multiple times without re-flipping the coin doesn't change the probability of flipping the coin, but it changes your probability of getting the answer right.
I agree
But probability depends on who is being asked and what he knows. You're just avoiding the question
@@giladkay3761 It also depends on exactly the question being asked. "What is the probability the coin came up heads" is obviously 50/50. "What is the probability that we woke you up to ask that question" is not 50/50.
It's entirely possible for me to have a fair coin and tell you "I will ask you how it came up only if it comes up tails." That doesn't change the probability of the coin flip, but it totally changes how you should answer "what do you think came up?"
@@darrennew8211 I think that is the confusing part. Because we assume the questions mean different things by the fact that different probabilities came out for each question, instead of logically and semantically defining the difference between them.
It doesn't change anything. The waking up occurs AFTER the coin toss. Saying anything that can happen after the coin toss has any bearing on the toss itself. violates the law of causation, because toss causes waking up.
Numberphile exposure has turned Brady into a bona fide mathematician, and I am firmly here for it.
This is more philosophy.
@@chessthecat It requires logical analysis and problem solving skills, which are related to mathematical skills.
The thirder and halfer arguments are talking about two completely different probabilities: the probability of the coin being heads *given* that Sleeping Beauty was woken up, and the probability of the coin just being heads. One is conditional, the other is not. Which goes back to what Tom was saying about it being about what we're really asking Sleeping Beauty. Are we asking about the conditional probability or the unconditional probability?
But the probability that sleeping beauty is woken is 100% though, so that doesn't change anything, she's going to be woken up anyway.
Exactly my thought, and I think that Tom saying that both views are supported by mathematicians is a little misleading. The real lesson shouldn't be that statistics depends on your opinion, but that statistics depends on the facts that you are given.
Nobody is asking a different question, it's just that the probability of the condition is 100% in any case, so it can be ignored.
@@ΠαναγιώτηςΓιόφτσος Yes, I honestly take offense to the statement that both are supported. One view is correct, the other view is wrong. There is no gray area here.
@@ΠαναγιώτηςΓιόφτσοςI disagree slightly on the last part, statistics depend on your interpretation of the question and the information you are given. The same information and question can be provided, but interpreted differently with different results.
And that is often used to twist statistics to show what the person wants them to show.
Have you also felt this discomfort when the needle goes into her head?
Won´t get nothing injected by a mathematician!!
Yes, the animation was horrifying 🤣
Yes, but I quickly forgot about it.
Can honestly say I never expected to be cast in the role of sleeping beauty in a Numberphile video…
Well it was more likely an outcome, than the thirder option being factually correct, after all no matter how infinitesimal the outcome of you being the sleeping beauty in a Numberphile video may be, let's say 1/TREE(100^10^3.1) that's still infinitely more likely than the thirder option which has 0 probability of being correct.
A coin was flipped, it had 50% chance of being heads, no matter how many times anyone wakes up to answer the question in the first place.
Is it inaccurate tho?
You mean you don't remember it.
@@livedandletdie Of course the coin comes up heads 50% of the time but think of it like this. Every time it comes up heads you are shown the coin once, every time it comes up tails you are shown it twice. The coin is still fair but picking a random viewing of the coin, what is the chance you see a tail?
Christina Aguilera sang it best.
Like Brady was getting at, it's not a mathematical paradox, but a language paradox because the question is vague enough that it can be interpreted as asking a simple question of the probability of a coin flip, or as asking the likelihood of waking up by a heads or tails. (There was a 66.7% chance you were woken up by a tail, but objectively only a 50% chance that a tail was flipped. Those are two different answers, assuming two different interpretations of the initial question, that don't actually contradict each other)
Except it is a mathematical problem, and that is that the precise question matters.
Mathematics without context is useless.
Mathematics is only useful because it relates to problems.
In this case how situations influence otherwise fixed probabilities
In this case, Mathematics and Language and linked.
There is actually a 50% chance that you were woken up by a tail.
And if it was tails, then there is a 50% chance it is Monday and a 50% change it is Tuesday.
If it was heads, then there is a 100% chance it is Monday.
So in total: Monday+Heads 50%, Monday+Tails 25%, Tuesday+Tail 25%.
That in the case of Monday+Tails you will also be woken up the next day makes no difference here.
That in the case of Tuesday+Tails you were also woken up the previous day makes no difference either.
@@ronald3836 lol, that doesn't make any sense. There's a 66% chance you were woken up by a tails. That part is out of the question. It's because if it's tails, you'll be woken up twice as often as when it's heads. It does matter that you'll be woken up twice for a tails, it matters very much.
The "paradox" is in asking "is the coinflip heads" vs "you were woken up: is it heads?". By rephrasing it you have solved the paradox and are now just getting the wrong answer :")
@@JoQeZzZ in the whole experiment, there is a 50% chance that you are woken once by a heads and there is a 50% chance that you are woken twice by a tails. This follows directly from the coin being fair and the description of the experiment.
In the tails case, you wake up twice. For one particular awakening, the chance of waking up on Monday is half the probability of the tails case, and the chance of waking up on Tuesday is the other half. So both have probability 25%.
I agree its not a paradox, theres a semantic switch up to conflate probabilities for two different things
yea I think on high levels of math like this, real paradoxes can't really exist anymore. There's Law of Excluded Middle stuff if you go into the depths of logic, but with things like this there's usually a sleight of hand or a subtle misunderstanding somewhere. If you've seen jan misali's video on paradoxes, he discusses how there are REAL logical paradoxes and linguistic/trickery/counterintuitivity paradoxes, and that's kinda the distinction I mean here.
I don’t get the thirder position tbh. Let’s even assume that one gets woken up 1000 or a million times when it’s T and only once when it’s H.
If I’m sleeping beauty and I get woken up and asked the question it seems like it’s 50% that I’m on the streak and I don’t remember (and will not remember) the other times of the streak and 50% chance this is the only day I’m asked.
(Although if I’m betting it’s obvious it’s better to bet on T assuming one wins an equal amount each time one is woken up. But I guess this changes if one maybe looses some proportional value each time one is wrong)
@@Nia-zq5jl Thinking in terms of betting is exactly the right way to reason about this. From SB's perspective, tails will be the winning bet 2/3 of the time. So 1/3 chance of heads is the right answer.
It's an important semantic argument though because it clarifies the definition of probability.
@@jonathanmooser6933 The halfer response to "If you are right we will give you 1000 pounds" would be "If you are right EVERY TIME WE WAKE YOU UP we will give you a thousand pounds at the end of the experiment".
Perhaps a middle ground offer would be "at the end of the experiment we will give you a thousand pounds divided by the number of times you are woken then multiplied by the number of correct answers you give". That way there are consequences for bad bets. I think this offer really leans towards the halfer worldview though because if you systematically chose heads or tails you would get the whole money half the time and no money half the time.
Even using probabilistic choices, flipping a coin each time you woke up to choose would lead to £500 on average. With a 6 sided dice we say tails on 1 though 4 or heads on 5 or 6. If it is tails we expect to win 2/3 of the money and heads we expect to win the money a third of the time. This leads to the expected winnings overall to be £500.
Ultimately the "winning 2/3 of the time vs winning 1/3 of the time" dilemma is more akin to "I'm going to flip a coin. If I flip tails and you guess it you get £2000, If I flip heads and you guess it you get £1000, if you guess wrong you get nothing. What do you think it will be?" The probability of the tails isn't higher but the expected winnings of guessing tails every time is twice as big as guessing heads every time
I love when a thought experiment is so strange that you also have to imagine that consent was given
At least it's better than those thought experiments about burning cats or putting babies in blenders. I'm not sure consent would help with tying people to a railroad track, though. XD
Imagine consent to put sleeping beauty to sleep is not requested, what is the probability the study makes it past an ethics board
Well, many of them are about prisioners in death row, for some reason
@@jakemetzger9115I'm not familiar with the thought experiment about babies in blenders. Jokes? Definitely. Thought experiments? not so much
@@TheDrinkingFood Depends on whether the ethics board is using a fair coin when making their decision.
The third argument is the probability per day. The half argument is the probability per experiment. They're measuring different probabilities.
Imagine the experiment is done 10 times, half the time it shows up as a head and half the time it shows up as a tail. And she will always guess a head. Well she will be correct in 10 days out of 30, but she will be correct in 5 experiments out of 10.
Thank you, your explanation made it more clear than the video did!
Well put
But the question was "what is the probability it was a heads" not "was it a heads"
@@TheCphase if you guess right half the times (5 times out of 10), then the probability was 1/2.
@@ronald3836 Correct, but the question is not at all about guessing right.
Brady killed it in this one! It was a very perspicacious way to avoid the paradox.
Okay Andrew
*Perspiration
*Precipitation
*Preparation
*Porridge
With the betting version, if she gets a payout every day, then the thirder stratergy works. If each day she is asked whether she wants to commit to a single bet when she is finally woken up, the halfer stratergy works.
Just consider the scenario where she's only woken up if it is a tail and not at all if it is a head. In that case, if she is being asked the question at all, then it means 100% that the experimenter has flipped a tail. Clearly the 1/3 answer is a posterior probability that is completely arbitrary and determined by experiment design. It is the probability that heads HAVE BEEN flipped, like the probability that it HAS rained if you see puddles on the street, which does not tell you anything about how likely it will rain at any given time.
Yes I had the same thought. It is kind of like the question, what is the probability that our universe supports human life. People may argue the probability is infinitesimal. But the only universe in which we can ask the question is one which supports human life. So the answer has to be one.
@@simonr6268too far bro
@@yoursleepparalysisdemon1828it is never too far
Yes, in that scenario the answer would be 0%. But that's not the scenario we're in. In our scenario it's 50%.
Seeing puddles in the street increases the likelihood that it rains and acts as evidence. Waking up does not increase the likelihood of tails and acts as zero evidence. Thus the posterior equals the prior.
I feel that there there are two 'probability spaces', one from an external observer and one from sleeping beauty, and these two are squished together. The events from the frame of reference of sleeping beauty is squashed into one half of the outside observer, and the outside observer's space is squashed into the 1/3rd of sleeping beauty.
I've a strong intuition this lesser-known paradox is tied to two infamous ones: the "Monty Hall problem", and the seemingly impossible "observer effect".
@@donweatherwax9318 Monty Hall isn't a paradox though, there is one and only one correct answer to monty-hall questions. This Sleeping Beauty Paradox has two valid answers,i.e. a true paradox proper.
@gregoryfenn1462 Granted; however, I'm not sure _this_ one is really a paradox. 'Paradoxes' have been solved, as Monty Hall has been; and I feel like this one may have a correct answer too. (And just as with Monty Hall, perhaps not the 'intuitive' one.)
Did you know, a poll was conducted of people's opinions. 50% were halfers, but 2/3 were thirders.
The confusion arises when you think of every “wake up” being independent, but they aren’t.
The 99 wake ups on T is 1 event and the 1 wake up on H is 1 event. There are only 2 events, not 100 events. The answer is 50/50 and the princess gains no more likelihood of being right for saying T. From her pov she only wakes up once and only has one guess.
"If you guess correctly, you gain $1." Clearly, this would lead you to guess T because the expected value is higher. But, really, that's just a different way of saying "If you guess T and are correct, you get $99, and if you guess H and are correct, you get $1."
@@renmaddox Now try other two variants:
1) If you guess wrong you loose $1 and if you guess correctly you gain nothing.
2) If you guess correctly you get $1 and if you guess wrong you loose 1$.
@@kgsws 1) Guess T and are wrong lose $1, guess H and are wrong lose $99. Still guess T
2) Is just the individual cases for gain/lose if you stick to the same answer each time. T and correct get $99, T and incorrect lose $1, H and correct get $1, H and incorrect lose $99
@@fantom789 Yes, you are right!
And each scenario always has only two unique outcomes.
@@kgsws The first of those doesn't really change anything meaningfully. It reverses the "right" choice, but for the same reasons.
The second is seems to be effectively the same as my version: Might as well guess H, because either you're correct and win $99, or you're wrong and lose $1.
You can make it more obvious by making two different games:
1) Give her 1 point each time she guesses correctly. This results in the thirder position.
2) Give her 1 point only if she answers correctly every time (note that since she has no recollection of having woken up before she will necessarily guess the same every time). This results in the halfer position.
You get woken up NOW. You're asked to give your best guess of the coin flip NOW. It seems quite unambiguous that the imaginary point will be given to you based on your answer NOW, and the value of your answer won't be halved if the coin landed on Tails.
@@eugenehertz5791 the ways to arrive at NOW is 1/2 via heads+Monday, 1/2x1/2=1/4 via tails+Monday, 1/2x1/2=1/4 via tails+Tuesday.
@@eugenehertz5791 terrible analysis. This is a new scenario with more detail added by Ben - it's "quite unambiguous" that this scenario wasn't presented in the original problem and there were no points given at all. Either get on board with the updated situation or ignore it, don't try and force your own third completely new situation and claim it's the "true one"
Nope. It's always a coin flip guess. Just because you might have been woken up repeatedly doesn't make it any likelier to have been heads, ever.
I disagree.
The first game will lead towards the best strategy being choosing at random. In the only monday case, you have a 50/50 chance of saying the right response, and in the multiples case, you on average will get half of your answers right.
The second game, best strategy is always say the same thing. (50/50 chance of being right.)
Asking her the question “is it heads or tails?” is different than asking “is her total score for correct vs. incorrect higher if she always answers heads or tails?” Then you realize obviously what she should say.
I think this is the closest insight in the comments. The problem has nothing to do with conditioning on "woken" but something to do with " the probability that a YES OR NO question being answered correctly" vs. "the probability of a coin flip".
I feel like that's just changing the question to something that has an easier answer.
@@meeharbin4205…and it happens to be an equivalent question
@@oldvlognewtricks I dont believe so
@@meeharbin4205 What exactly makes it a different question?
Absolutely love Tom's enthusiasm, all the videos featuring him are a treat (even for a topic as infuriating as that one!)
Your explanation was FAR better than Veritasium's, I finally understand what's the paradoxical part about this. It's a fun sleight of hand! Of course the probability that the coin comes up heads remains 1/2 no matter what, but if that's true then why does the calculation say 1/3?? Because the assumption that these three equal probabilities have to add up to 1 is false! These are just the cases where SB is woken up, which is 3/4 cases, so that's what the probabilities add up to. If you add up the case where the coin came up heads on a tuesday (which means SB is not woken up), THEN you get to 1. And yes it's the same as the other three again, so we get 1/4 for all of them, and properly 1/2 for all the Heads cases combined.
So indeed the difference is if we're asking about the coin's probability distribution in general (which never changes from P(H) = 1/2), or about what the coin came up as THIS TIME. Which, because SB is awake, is conditioned on the fact that SB was woken up, and the answer is therefore P(H | SB woken up) = 1/3.
Your explanation here makes a lot of sense, but having watched the video I've no idea how you got that from it tbh.
@@MrDannyDetail honestly agreed, I wish this explanation was in the video rather than in a comment by someone else. The part I meant that was better here, is just the fact that I'm taking the thirder position seriously now. I don't remember exactly, but at the time I don't think Veritasium did the thing of "proving that all three wake events have the same probability"... he probably just said it, and my response was "wait wtf no they don't!" -because they don't unless you do the "conditioning on SB waking up" trick- ok they do, that was actually a mistake I made at the time. But their probability without the conditioning trick would be 1/4 and not 1/3, so the point still stands.
Anyway, the fact that the calculation was done here and naturally gave 1/3 blew my mind. And now that I understand how it's actually a valid calculation that answers a sensible question, that has really expanded my perspective on this.
The question asks about what event happened when the coin was flipped (t or h). When the question is repeated on tuesday it doesnt change that there was only one event that could lead to that. So when they list out P(mon n T) and P(tue n T) they are equal because they are the probability of the same event. Listing both out double counts them and if you count them just once the paradox goes away.
I think there could be two interpretation to the problem:
1. The probability of coin flip being a tail.
2. The probability of you being correct if you answer tail.
These are two different questions and the probabilities are different.(to be honest I feel like this is an ill-defined question)
I think you are the only person in the comment section who solved the paradox. Strange Numberphile didn't 🤔
I agree with Brady. You do different things AFTER the coin was either a tail (50% chance) or a head (50% chance). What you are asking isn't the probability the coin would land on tail or head, but what is the probability of a COMBINATION of things (one thing being tossing the coin, and the other thing being the different amount of time you will then be awaken). As Brady said, she is given more chance to be correct when it's a tail.
Back in college I did a project about the Brier score, which can explain the probabilities used in weather forecasting as an example.
If we ask Sleeping Beauty the question every time we wake her up, she would minimize her Brier score by saying 1/3. If we ask her the question once during the experiment, it is minimized with 1/2.
Since the question as stated was that "one of the questions she is asked is...", I'm more on the 1/3 side, but strongly understand the argument for 1/2!
Oh I didn’t finish the video :D this is even brought up
There is one thing that wasn't mentioned which is, imo, the deciding piece of information. When sleeping beauty is asked the question, there is one piece of information that she knows that the amnesia drug can't take away from her: the fact that she has been woken up. Overall, there is a 1 in 4 chance that each of the following options will happen: It's Monday, the coin is heads, and she has been woken up; It's Tuesday, the coin is heads, and she stays asleep; It's Monday, the coin is tails, and she has been woken up; and it's Tuesday, the coin is tails, and she has been woken up. All of those options are equally likely and the coin flip is 50/50. But given that she has been woken up, that eliminates the option that it is Tuesday and the coin is heads (because she would still be asleep). The remaining three options are still equally likely. Therefore, given that she has been awakened, there is a 2/3rds chance that the coin was tails and a 1/3rd chance the coin was heads.
This actually reminded me a lot of the Monty Hall problem and how 2/3rds of the time if you switch, you'll get the prize. Although the math is different.
"All those options are equally likely" is what's in doubt. If the coin landed on Heads, then her wake up must be the Monday, so 50% chance of Head+Monday. If the coin landed on Tails, then the wake up could be Monday or Tuesday, each with equal probability. So there's 25% chance of Tails+Monday and 25% chance of Tails+Tuesday.
The question is the probability of the flip, not a request for Sleeping Beauty to guess what the result of the coin flip was more often or some other similar question (such as what day is it). As stated in the video, the situation (and available information) never actually changes as it does in the Monty Hall problem.
There is actually an infinite number of events that aren't part of the experiment that you can't count. You can't include heads on a Tuesday in your math because it isn't a scenario that can happen and everyone knows it.
This reminds me of the anthropic principle. Our universe may be unlikely not by random chance but because in a likely universe nothing would be able to observe it.
Actually the sleeping beauty problem was invented by Arnold Zuboff and named by Adam Elga. The original formulation of the sleeping beauty problem was also about the antrophic principe. The question is whether or not your experience is itself evidence for something.
Please read Arnold Zuboff's important work on the sleeping beauty problem.
Brady's wagering argument is, I think, the key to this whole problem, or at least it's what unlocked it for me: What's changing isn't the probability, it's the odds. If you asked her to bet a dollar on the result of the coin every time she was woken up, then if she always guesses heads she's risking $2 to win $1, whereas if she always guesses tails, she's risking $1 to win $2. That makes guessing tails the dominant strategy, even if the coin itself is a fair 50/50 shot. The 1/3rd argument, then, isn't really asking how likely the coin was to be heads, it's asking you for the payout you'd expect if you assumed it was.
The two different answers are optimising different outcomes. 1/2 is the answer to give if you want to minimise the average number of times you are wrong _this one time the question was asked_ . n/(n+1) is the answer if you want to minimise, on average, how many times you were wrong _over the duration of the whole experiment_ . In both cases the average is over many copies of the whole experiment done iid. The two are subtly different, which shows that probability (because it's a degree of belief) is not only subjective but context dependent.
There is no optimising to be done. The is also no "being wrong", the answer is always the same, since the probability of the coin toss does not depend on how many times she wakes and thus is a constant number: 1/2
Yes I think this is right. There’s two different questions you could be asking.
I think this is precisely the point.
Exactly what I was thinking. The seeming paradox derives from ambiguity in language of the question posed to the sleeper such that they could answer in a manner that focuses on their true in-the-moment state of evidence (“halfer mindset”), or they could answer in a manner that employs a decision theory framework in combination with current evidence state. As usually formulated (including in the video) the cost/benefit matrix for use in the decision theory framework is itself left nebulous, but humans tend to like the feeling of being “correct”, hence the strategic dominance of a guess-tails strategy manifest by the “thirder” mindset.
You can also get 1/(n+1) instead of n/(n+1) if you want to maximize the chance of being right at least once over the course of the experiment. 😀
I have seen this "story" before, and so long as the question is "what is the probability that the coin flip was tails", I have the same answer now: the probability is 50%, no doubt about it.
Waking up one time or two times doesn't play any part in the probability because the two events are based on the same coin flip, whether or not she will be woken up in the future, or was woken up in the past plays no role in the probability.
If the question was instead, "what is the probability that you have been woken up today because the coin flip was tails" then the question changes, and the answer is 1/3rd.
Correct. Not sure why this is hard for some people to the point where anyone would label it as a "paradox".
But no matter what the coin flip was you'll be woken up, so both statements are equivalent
Ultimately this is all semantics and you've already stated the "intellectually honest" way to ask the question, but the simple inclusion of past tense in the way the question is asked is enough to change those semantics, at least in my opinion. The experiment is fully explained to you beforehand, you are aware that a coin flip happened and the result has been recorded.
In effect, simply being awake indicates one of two things: It is not a Tuesday, or the coin came up Tails.
In other words, 1/4th of the time when you are 'not woken up' is functionally the same as being woken up and shown that the coin came up Heads. If you were asked "what is the probability that the coin flip was tails" while actively being shown that the result was Heads, surely your answer would be "0%"?
In the end this whole thing just seems like a clever way of restating the Monty Hall problem. By being awake, you are basically being shown what's behind "door number 3".
@@giladkay3761 if you wake up, there's a chance that it's Tuesday, and the probability that the coin flip was tails is zero in that case. If you're asked about "waking today in particular" the answer changes compared to being asked about "waking at any time"
@@Hyatice no, in this problem you are NOT given any information by waking up. You knew already that you would wake up, whether coins or tails. Therefore nothing changes and the probabilities after waking up are still 1/2 and 1/2.
If heads, then you will wake up on Monday. This is clear. Half the time you run the experiment this will happen.
If tails, which happens half the time, then you wake up twice, but both of those times you are not aware of the other time. So you can't tell if this is the first time or the second time you wake up. Both are equally possible. Probability that you NOW woke up on Monday is 1/2x1/2 and that you NOW woke up on Tuesday is 1/2x1/2.
So if you wake up, the probabilities are:
P:(H on M) = 1/2, P(T on M) = 1/4, P(T on T) = 1/4.
P(H) = P(T) = 1/2
P(Mon) = 3/4, P(Tue)=1/4
It's either going to be a Monday or it's going to be a Tuesday when she wakes up. It's 1 of 2 situations for her. I think the paradox is created because, using the illustrations in the video, the fourth situation in the bottom right square where heads was flipped, it's Tuesday but she is not awoken is not talked about but it is still an event that has a probability of happening. But it seemingly doesn't "exist" or happen for her because she is not conscious.
This feels basically like nonsense. If both answers are "valid" and "accepted" (tbh kind of makes me cringe to hear a mathematician say that!) then the question is not well formed or the wrong tools are being applied.
The question has multiple interpretations and therefore multiple valid and accepted answers. It's a type of paradox, rather than some kind of mathematical statement, no nonsense going on here
Right, I don’t understand how it is a paradox.
There is a bayhsian updating here. There are not 3 situations, there are 4 with equal probability. You're missing the scenario where it is Tuesday and she isn't woken up. When she is woken up, she knows that possibility is eliminated, and so her probabilities update, knocking it from 4 equal probabilities down to 3.
This is one of the few numberphiles I've had to re-watch for comprehension. I LOVE the brain-strain here. I've got a variation for you to play Sleeping Beauty:
"Sleeping Beauty is going to be put to sleep on Sunday. The experiment hosts will flip a coin after she goes to sleep and if it comes up heads, she will be woken up on Monday and asked a question and be put back to sleep with the memory of Monday wiped from her mind. if the coin comes up tails, she will be woken up on Monday and follow the procedure as if it were heads, however she will be woken up again on Tuesday and repeat the Monday procedure again, wiping the Tuesday from her mind. (all the same as the original experiment) She'll be woken up on Wednesday to end the experiment.
However in this variation, she is given a coin which she is allowed to flip to help her acheive a 50% randomness to aid her answer, but she's only allowed to flip it once and must choose to either flip it before going to sleep on Sunday, or she will flip it upon being woken up on Monday as well as on Tuesday. She will not know on Tuesday that she already flipped the coin and will be able to choose to flip the coin on Tuesday as if she hadn't woken up on Monday.
The question she will be asked each day she's woken up "Did the coin flipped on Sunday come up Heads or Tails." And she will be incentiviced to have answered correctly the most amount of times. This begs the strategy... Should sleeping beauty flip the coin before going to bed and always answer according to that result, or should she flip the coin after waking up and answer according to that result."
It's a weird variation that I think plays at the question of stuborness vs flexability in the face of many unknowns.
This is deeper than I thought, I've flipped twice but now I'm confident it is 1/3 because if you keep score of his answers then that's the answer that will tally correctly over the long run.
PS. I looked at wikipedia where the question is "What is your credence..." which I think is subtly different to "What is the probability..." . It clarifies that things are to be looked at from the sleepers perspective.
Seems to me this is simply an issue with the ambiguity of language.
Sleeping Beauty is not asked, "When this fair coin is flipped, what is the probability it would be tails?" But that's what we usually think is meant. Instead, she is asked, "Given _this experimental setup,_ what is the probability that we flipped this fair coin and got tails?" The two questions seem like they should be the same, but they aren't. The first question entails its own answer, as it is a fair coin. The second question is about the experimental design. In the Rip van Winkle extension, if Sleeping Beauty were to repeat this process (let us assume it is hours instead of days, simply to spare her the time!), then she will sleep for just about six weeks (41.666... days). We can then consider the experiment repeated for a year (about 8 times). On average, we would expect 4 heads and 4 tails--which means about 4 awakenings after 1 hour and then sleeping for 999 hours, and 4000 awakenings spread across every hour of the ~6 weeks. Even if she got a string of heads, such that the coin only came up tails a single time, the tails awakenings would still vastly outweigh the heads awakenings, e.g. 7 heads-awakenings vs 1000 tails-awakenings, despite that event being quite unlikely (3.125%). Hence, she should consistently bet on getting a tails-awakening, rather than a heads-awakening.
Perhaps that's the best way to look at this question. It _sounds_ like we're being asked what the probability of the coin is, but we aren't. We are being asked what the probability of getting a heads-awakening is vs the probability of getting a tails-awakening.
Consider what I will call the "memory wipe" variant. Sleeping Beauty wakes up, unsure of where she is. She is then told that she has just finished the experiment; it is now Wednesday and she is free to go. But, before she leaves, she is asked what her credence is for whether the coin was heads or tails. The only information she has gained is that she is done with the experiment--this awakening _is not_ contingent on the coin. Quite obviously, her credence must be 1/2, as she has not yet learned which path she took. _This_ question has returned to being about the coin itself--because, to use Brady's terms, she is no longer walking the path. While she is still _on_ the path, the question is about the path, not the coin.
Did you just ask chatgpt to hallucinate a solution for you? You added some fluff to that explanation that makes no sense!
In all that text and baffling conversations of hours into years for no reason, you didn't explain why you would read the question the way you did. How can you see "how many awakenings happen on a tail flip" from the question as asked "what is the probability the coin was a head?"
None of the permutations of the question in this video come close to what you seem to be assuming.
But this is what makes me wonder if there is some version of the paradox out there that actually does ask a sufficiently ambiguous question to seem paradoxical, and this version just messed up the question a bit.
I genuinely love the visceral GLEE on Tom’s face through this.
Here's my opinion as a mathematician: the situation is analogous to the difference between asking someone about the probability of the coin landing heads without showing them the outcome, and the same question after showing them that it landed tails, i.e. the difference between unconditional & conditional probabilities. The conditional information in the sleeping beauty problem (i.e. the fact that she has just woken up) is less strong than knowing the actual outcome of the coin flip, but still enough to make the probability quite different from the unconditional probability (i.e. 1/3 instead of 1/2). If we allow sleeping beauty to wait a couple of days before answering, then if she isn't put to sleep again she should answer 1/2.
If sleeping beauty is told (after waking up) that she can win some money if she correctly guesses the coin flip then she is more likely to win if she says tails than heads, i.e. if we repeat the experiment multiple times with 2 different sleeping beauties; one of whom guesses tails and one who guesses heads, then the one who guesses tails will win more money.
All those arguments about saying that the probability of the coin landing heads is always 1/2 regardless of her situation are disregarding the extra information that she has from knowing that she has just woken up, and so it's a bit like flipping a coin, seeing the outcome is tails, and then saying that the probability that a head was flipped is 1/2.
The correct question is: what is the conditional probability of heads given that SB is woken up. Let us calculate this:
1. Since the coin is fair, P(heads) = 1/2.
2..SB is always woken up, whether heads or tails. So P(SB woken up) = 1.
3. Since SB is always woken up after heads, the event "heads AND SB woken up" is the same as "heads", and P(heads AND SB woken up) = P(heads).
Thus:
P(heads | SB woken up) = P(heads AND SB woken up) / P(SB woken up) = P(heads) / 1 = 1/2.
So 1/2 is the correct answer, being the conditional probability of heads given SB is woken up.
@@viliml2763 you haven't accounted for the day (which is random from SB's perspective)... in fact the probability of being woken up on a particular morning during the 2 day experiment is different depending on the coin flip; P(woken up|heads) = 1/2, P(woken up| tails) = 1.
To be more explicit; P(woken up | heads) = P(woken up & 1st day | heads) + P(woken up & 2nd day | heads) = P(1st day | heads) + 0 = 1/2 + 0 = 1/2
P(woken up | tails) = P(woken up & 1st day | tails) + P(woken up & 2nd day | tails) = P(1st day | tails) + P(2nd day | tails) = 1/2 + 1/2 = 1
@@viliml2763 Well, if you're asking if waking up makes a fair coin into a not fair coin: obviously it doesn't. But the thing is that here (and in the video, since it seems to be ambiguous) P(heads) is not well defined. This is where the "paradox" arises.
If P(heads) means "the probability that a fair coin comes up heads" you are absolutely correct and the answer is 1/2, regardless how often SB wakes up.
But P(heads) can also be interpreted as "the probability SB sees at the moment for the coin having come up heads on Sunday", which would mean that you can't automatically assume 1/2 for P(heads). That in turn leads by lack of information to the need for another assumption, and the most popular one is "since we cannot distinguish the instances of waking up, let's assume they're equally likely", which like it's shown in the video leads to 1/3 as the answer.
So as long as the question is not making clear what exactly it is we want to know, both answers can be argued to be correct.
@@viliml2763 Except that the conditional information isn't "SB was woken up at least once," it's "SB was woken up right now." "Right now" might be Monday or Tuesday, which changes the conditional probability of "SB woken up."
Since time moves forward independent of our coin toss, P(heads AND Monday) = P(heads AND Tuesday) = P(tails AND Monday) = P(tails AND Tuesday) = 1/4.
P(SB woken up) = P(SB woken up | Monday) * P(Monday) + P(SB woken up | Tuesday) * P(Tuesday) = 1 * 1/2 + 1/2 * 1/2 = 3/4
Our question is, as you said, P(heads | SB woken up) = P(heads AND SB woken up) / P(SB woken up)
= P(heads AND Monday) / (3/4)
= (1/4) / (3/4)
= 1/3
When the coin is flipped, there's a 1/2 chance that the coin lands on heads and a 1/2 chance that it lands on tails. Therefore, if the experiment is repeated multiple times, then the number of experiments in which the coin landed on heads is equal on average to the number of experiments in which it landed on tails. The simplest version of this is that the experiment is done two times, with the coin landing on heads one time and the coin landing on tails the other time. Over the course of both experiments, she'll be woken up a total of 3 times. Two of the three times she's woken up, the coin would have landed on tails, while one of the three times she's woken up, the coin landed on heads. If she's asked what the coin landed on each time she's woken up, she should say 'tails' two of the three times and 'heads' the other time if she wants to get all 3 right. Therefore, the probability that the coin landed on tails on any given time that she's woken up is 2/3.
Pretty easy, but a cool puzzle! I liked this one.
A lot of people reverse the conditions for her being woken Monday and Tuesday, I'm going with she is woken on Tuesday if it comes up tails and not woken on Tuesday if it comes up heads.
Here is how I would motivate the paradox. So usually when you bet you assign your bets in preference to your sense of the odds (probability) therefore a bet should reflect the probability estimate.
So if as Brady suggested (and occurred to me as I was watching) we include a payout related to her answer to the question "What do you think the coin came up?" asked when you wake her up on Monday and Tuesday with her getting $1 if she correctly guesses and nothing if she is wrong.
In such a scenario the expected payouts for different strategies are easy to specify. If Sleeping Beauty absolutely favours heads then we expect her to get 50 cents (half the time $1 and half the time nothing). If she is indifferent to what she bets and before answering flips her own coin then we expect her to win 75 cents (this is a bit tricky to show but I think I got the combinatorics right). Finally if she absolutely favours tails we expect her to get $1 (half the time she gets $2 and half the time nothing).
We can even turn it into a non-monetary bet, let's say Sleeping Beauty finds it very fun and exhilarating to be shown video footage of her being correct in a guess. The experimenters video tape all her Q&A sessions. She will be shown them on Wednesday. So on Sunday she knows if she commits to betting Tails she will either see one piece of footage where she wrongly guesses tails when it was head or two pieces of footage when it was tails (and she got it correct), so she commits to tails since it is the way to get the best chance of the most video of her winning. We might say she is maximizing her chances of being right by preferring tails (as if tails was somehow more probable), but it probably needs to be more precisely said she is maximizing her emotional elation at being right in the same sort of true stimulus/recording being played.
Given that her winning strategy is guessing tails does that mean she has estimated the probability of tails as absolutely greater (and given the expected payouts she favours tails 2 to 1). If we really only laid our bets according to our understanding of the odds than yes.
However my initial statement missed something betting reflects the odds all other things being equal. If you have different payouts for different bets that changes ones betting strategy or different numbers of winning opportunities. This is the entire reason that odds determine payouts, the booky gets people to bet on the long odds by having proportionally large payouts.
If you can buy one of two kinds of tickets and type A has twice the payout of type B if it wins but the probability of A wining is the same as B you favour A. If you can buy twice as many type A tickets as B but the odds and payout per ticket is the same whichever you do, you buy twice as many chances to win (twice as many As) and so on.
Regarding my idea of showing Sleeping Beauty video of her answers to elicit an emotional response. What if she is bored by having to see the same right answer twice, but it actually lessons her disappointment by acclimitization to see herself get the same wrong answer twice. Then she should favour answering heads. As it will be more agreeable to watcher herself get heads right once if she guesses that and it turns out right than the tedium of watching her get tails twice if its tails and she guesses that. Conversely it will be easier to take getting tails wrong by saying heads and seeing herself do the same wrong thing twice where the familiarity softens the blow to the sharp shock of guessing tails and seeing herself get it wrong only once.
Edit: Likewise you can imagine she has emotional responses such that she enjoys the surprise of giving different answers if it lands Tails (one correct, one incorrect), so flipping a coin would be favoured if she finds that more enjoyable than two correct answers and for heads finds the 50% chance of correct balanced by 50% risk of incorrect. So depending on what objective facts you are tracking and what measure of correct you have lots of different answers.
reply for future refrence
For the "thirder" argument it is asserted that: P(T|Mon) = P(H|Mon) but if you work out the relationship assuming a fair coin you get that P(T|Mon) = 1/2 P(H|Mon) which would mean that:
P(T∩Mon) = 1/4
P(T∩Tue) = 1/4
P(H∩Mon) = 1/2
P(H∩Tue) = 0
therefore:
P(H) = P(T) = 1/2
See full reasoning below:
The coin is fair:
P(T) = P(H)
P(T) = P(T∩Mon) + P(T∩Tue)
P(H) = P(H∩Mon) + P(H∩Tue)
Since (H∩Tue) does not happen:
P(H∩Tue) = 0
Since you can't tell which day it is if it is tails:
P(T∩Mon) = P(T∩Tue)
Therefore:
P(T∩Mon) + P(T∩Tue) = P(H∩Mon) + P(H∩Tue)
P(T∩Mon) + P(T∩Mon) = P(H∩Mon) + 0
2P(T∩Mon) = P(H∩Mon)
Since P(A∩B) = P(A|B)P(B):
2P(T|Mon)P(Mon) = P(H|Mon)P(Mon)
P(T|Mon) = 1/2 P(H|Mon)
I believe the discrepancy here comes from a lack of clarity: in that section P(T) and P(H) should not imply P("a fair coin toss comes up heads/tails") respectively, but P("I got woken up just now because the coin toss came up heads/tails on Sunday"). - which is absolutely not clear from the notation and I only got that idea from what I understood about the context. But with that you can't automatically assume P(H) = P(T) = 1/2 and that with no other information one would guess P(T|Mon) = P(H|Mon) seems reasonable to me.
It’s 50/50. The amount of times you wake her up doesn’t affect the probability. I understand why people think this is a paradox, but it just isn’t. Yes, when you wake beauty up it could be any of the three possible wake up times (two on Monday, one on Tuesday), but two of those wake up times share the same 50% chance of the initial equation.
Incorrect. You seem to believe the question is "what is the probability of tossing heads".
@@메로이-f2l The original (a priori) is not what was asked for. The answer is 1/3.
I think the problem though is that Monday tails and Tuesday tails are not separate events because they are a result of the same event (one coin flip) that just spans two days long (two wake ups instead of one) and I think it's a fallacy to refer to Monday tails and Tuesday tails as separate events.
Tom could be just doodling nonsense on the paper and he would come up with a way of making it interesting. He just has so much passion for what he is doing :)
As with most of these probability paradoxes, it all boils down to understanding/specifying further exactly what the question being asked is
12:00 There is no difference between the coin flip vs. "the pathway that resulted from the coin flip". If the coin is heads then you end up on the heads path, and you only end up on the heads path if the coin is heads. The flip result and the path are equivalent.
So what are you claiming is the answer?
@@godfreypigott that's a separate question... and the whole point of the video is that 2 different probability frameworks give 2 different answers
@@5h5hz Except that the claim that the question can be interpreted in two different ways is incorrect. Only 1/3 is correct the way the question was phrased.
@@godfreypigott spoken like a true thirder
@@5h5hz
Run the experiment 100 times, 50 heads, 50 tails.
Sleeping Beauty will be woken 150 times.
50 times Monday following heads, 50 times Monday following tails, 50 times Tuesday following tails.
P (Heads) = 50/150 = 1/3.
This feels like it could rdebunk the hypothesis we're living in a simulation.
I think the best way to explain the problem here is the fact that what sleeping beauty is trying to do while answering is not fully defined.
Its basicly a question of how sleeping beauty measures her own success. Aka what her goal is.
If her goal is to be right the maximum amount of times she will answer one way.
If her goal is to be right at least once, she will answer with 50/50.
As with a vast majority of "paradoxes" it is simply the fact that the problem isnt defined enough. Resulting in multiple answers depending on how you fill in the missing information.
Of course the video describes changing what is being asked to fully constrain the problem.
But I think changing sleeping beauty's goals is a much more obvious proof that the problem isnt defined enough. Since the wording of the question doesnt need to be changed in order to explain it.
it's either ½ or ⅔ depending on your personal interpenetration of quantum mechanics
The "Sleeping Beauty Paradox" is such a mind-bending concept, and this Numberphile video does a fantastic job of explaining it in a clear and engaging way. It's one of those philosophical puzzles that can really make you question your intuitions about probability and decision-making. The discussion and different viewpoints presented here add an extra layer of intrigue to the paradox. Numberphile consistently delivers thought-provoking content, and this video is no exception. It's a great reminder of how math and philosophy can intersect in the most perplexing and fascinating ways.
I think the paradox is in the question. The question is asked 4 times, two on Mon and Tue (in Tail case), and two on Mon and Tue (while asleep) (in Heads case). So if each has 1/4 a chance, head/tails is 50/50, at thesame time 1/4 Head ((while awake)) is 1/3 of the 3/4 ((of questions while awake)).
Hey! Brady's a Dad! Congratulations
Personally I thought it's 1/3 at first but then I gradually convinced myself it's 1/2. Because it's like knowing beforehand that she would wake up in one of 2 universes - the one in which the heads dropped or the one in which the tails dropped and the number of wake-ups doesn't matter
@pinkraven4402 Correct!
I agree. Thirders are full of themselves if they think that a perfect coin toss changes its probability based on the number of times you check the result.
This was my train of thought but I couldn’t describe it as cohesively so thanks for this😂
What if I bet you $100 every time I woke you up?
That's a matter of number of wake-ups, not the probability itself. If I would pay you 100 times as much if your roll 2 sixes on dice than I would for any other result, it doesn't change the probability of double six being 1/36. Neither would paying it x times.
Halfers answer as a third-party perspective of the coin flip.
Thirders answer from Sleeping Beauty's perspective where the coin flip doesn't exist half the time for a heads result.
As multiple peope here and in video already said: its not really a paradox, its just a an interpretable question. The good thing about probablistic questions: we can just simulate them.
And when you ask the sleeping brady what the probability for Tails is on the flip on saturday, the answer is 50%. When you ask predict the flip on saturday the sleeping brady is correct 2/3 of the times with the answer "Tails".
And thats just because you ask him more often in the "Tails" case.
I really dont get why this should count as a paradox. There is nothing "paradoxic" about it, both cases are pretty clear cut. Ambiguous questions dont gerenate paradoxe. And its not even called a "paradox" on wikipedia.
I feel like those assumptions under the thirder argument are what mess it all up.
I prefer Brady's explanation: it's a 50/50 chance that the coin was a heads, but it's a 1 in 3 chance that she's in that specific situation.
If you'd like to disagree with the laws of probability then you are free to make your own mathematical model
Why do you think the assumptions "mess it up"? Consider the two possible interpretations of the ambiguous question:
Asking "is this fair coin fair"... is a ridiculous question with a trivial answer.
Asking "how would you bet on the outcome" ... makes use of all the effort gone to running the experiment.
Brady hit on it and it's something I've said before: if mathematics is the science of explanation, then probability is the art of asking questions.
So great to have Tom Rocks Math[s] Tom back in an episode of Numberphile!
This one feels like a cross between the Monty Hall Problem (referenced briefly in this video) and the Prisoners' Dilemma
In the original formulation of the sleeping beauty problem, The Monty Hall Problem is of relevance. The Problem was invented by Arnold Zuboff and named by Adam Elga. Please read His Important work.
I love this problem. It's both intuitive and easily provable that the probability that any coin on any day must be equal probability to any other coin and day, but the answer to the question depends entirely on whether Sleeping Beauty understands the whole experiment when you ask and what specifically you ask her.
Except that P(H on Mon) = 1/2, P(T on Mon) = 1/4, P(T on Tue) = 1/4.
That in the tails case you are asked the question twice does not mean it is more likely that the coin came up with tails.
In the tails case, when you wake up to Mon or Tue it does not matter that you are also woken up on Tue or Mon. You can basically ignore the other waking up. If it happened yesterday, you don't remember it anyway. If it happens tomorrow, you won't remember today anyway.
This is very interesting and I think it is very simply resolved by specifiying the question in more detail.
1. What is the likeliness the coin flipped heads? : It is 1/2
2. Considering you just woke up, what is the likeliness the coin flipped heads?: 1/3
It is way more fun and paradoxical if you just ask the 1st question.
Brady, you nailed it brother. The correct approach depends on what question sleeping beauty is asked.
1. Was the coin flip heads or tails? Halfers are correct.
2. What day of the week is it? Thirders are correct.
Cool. The action of waking up is transforming information. Consider sleep as 0 and wake as 1. This is what changes the outcome. So, if we write a truth table and consider the time she was asleep and when she woke, the probability is 1/2. Therefore, saying that she has no information is false because, as observers, we know that she both slept and woke up, which provides us with information.
Imagine being in a scenario where you're flipping coins in a room. By using the status of the coin to turn on/off the light(0,1), you observe 1 head (H) and 2 tails (T). However, you're deliberately excluding the times when it's heads half of the time. This intriguingly means there is no randomness involved. Here's the kicker: you have the ability to force the outcome to be any desired proportion. It's fully controlled . Consequently, you end up observing heads (H) one-third of the time and tails (T) two-thirds of the time. Nevertheless, this doesn't alter the probability of the coin itself, which remains 1/2 throughout.
The probability is 50:50 since P(T|Mon) != P(H|Mon). There are no reason for these two probability should be equal and P(T|Tue)=1!= P(H|Tue)=0. Interesting mistake that people would make is that asking multiple times duplicate the number of cases. But actually P(T&Mon) = 1/4 and P(H&Mon) =1/2
To me this feels a lot like that WWTBAM meme question, What is the chance you get the correct answer if you randomly choose an answer? A 50%, B 25%, C 25% or D 75%.
Oh, well done. You've just made my head explode. How am I going to clear us this mess? Thanks a bunch.
The issue here is that there are two different meanings for the word probability. Before the experiment the coin has a 50/50 chance of landing on either side, and there is no real fact of the matter. But when sleeping beauty is woken up the flip has already happened, so it must have landed on one side or the other and it doesn't really make sense to talk about the probability. Instead the question is asking about sleeping beauty's best guess of what happened given her knowledge of the situation, which gives the 1/3 answer.
Of course you could insist that the question is really about the initial 1/2 probability, but then there would be no point in the whole sleeping beauty story since whatever you plan to do arter the flip obviously has no impact on the probability before the flip. So its pretty clear that isn't what the question is going for, especially with the original "credence" phrasing.
YES. I'm so proud of Brady for calling out the semantics game that's being played here
Ok, now, imagine the sleeping beauty's friend experiment: you perform the sleeping beauty experiment to conclusion as usual and then ask her friend the probability she was woken up Tuesday during the experiment. As Brady points, asking her multiple times is what causes the conflicting answers.
Asking multiple times only matters if you ask about the expected number of correct answers. We're asking about the probability so it's equivalent to only asking once.
What a great video! I might be totally wrong, but it brings to my mind things like the anthropic principle, doomsday argument and Boltzmann brain problem. While I know the objective probability of heads and tails is 50/50, I can't help but find myself a thirder here!
what are those?
@@maxonmendel5757 Anthropic principle : if I'm alive, then the universe has to be just right for my existence to be possible (which doesn't mean it was fine tuned).
Doomsday argument : if I'm alive, assuming I'm one of the humans in the "most populated century" since that's the most probable century to be born in, then humanity should go extinct soon (but one has to remember that this is based on an assumption so can hopefully be wrong)
Boltzmann Brain : if I'm alive, it's very probable that I'm just a brain that popped into existence by random fluctuations and is generating random inputs that happen to make me experience a fake reality (but although it's more probable, reality seems fun so let's go with it)
@@ghislainbugnicourt3709 Huh, the version of the doomsday paradox I’m more familiar with is, “there a (e.g.) 90% probability that I’m in the middle 90% of all-humans-who-will-ever-be-born (ordered by date of birth), and therefore the number of humans who will be born after me is less than 95% of them, and so there will be at most 20 times as many humans born in the future as were born before me”
But, same general idea...
@@ghislainbugnicourt3709 wow thank you
I totally agree... the objective, external reality is that there is an equal probability of heads and tails, but the subjective reality is that it's more likely you're in a scenario where the coin toss result was tails.
I'm so glad you concluded "it depends on what actual question you're asking" because I was yelling that for the whole video.
And I think the two questions are: "What are the odds that the coin came up heads?" vs "what are the odds that this is the first time you've been woken up?" (first can be replaced by any number, but 1st is 0.2% and every other number is 0.1%). That's the only question that actually has a thousand possible options to choose from.
Tom derived one equality from one assumption, then another equality from a different assumption, then because one side from each equality was the same statement he then combined them into a threeway equality, but surely you cannot do that if each of the two equalities required a different assumption, because now the threeway equality surely requires both assumptions to be true or else it isn't valid, and making both assumptions simultaneously would sort of collapse the problem into a certainty that it was Tails and a Monday wake up anyway.
He removed the assumption by going from the conditional probability to the probabilty that both events happen.
Where the argument goes wrong is in P(T|Mon) = P(H|Mon). If you consider being woken up as the event, then if you know you woken up on Monday, it is more likely to be heads (2/3) than tails (1/3).
@@ronald3836 If you were woken up and told it was Monday (aka you know it is Monday) then surely the probaility is 50/50 for Heads or Tails, because the additional Tuesday wake up for the Tails path wouldn't come into it
@@MrDannyDetail No, there are 3 cases: H on Mon, T on Mon, T on Tue. The probability of H on Mon is 1/2. The probabilities of T on Mon and of T on Tue are both 1/4.
This means that P(Monday) = 1/2+1/4 = 3/4, P(Tuesday) = 1/4.
And P(T|Mon) = P(T and Mon)/P(Mon) = (1/4) / (3/4) = 1/3.
P(H|Mon) = P(H and Mon)/P(Mon) = 2/3.
What tricks the mind is that you are woken up both on Monday and on Tuesday in the tails case, but since you have amnesia there is no difference if you change that rule to being woken up either on Monday or on Tuesday with equal probability. As the video correctly stated, given that it is tails, the probability of it being Monday is equal to the probability of it being Tuesday, so both 1/4.
I think the best way to visualize this is with a tree. Imagine there is an apple tree with a distinct shape, its trunk splits in half, forming the shape o a “Y”. On the right trunk there are 10 apples growing and on the left trunk there are 20 apples. If you pick one of the apples, what is the probability of it coming from the right/left trunk?
This mimics the 1:2 ratio of SBs problem (Right = Heads/Left=Tails) but this way it's clear why there are two answers, it depends on HOW you chose to pick the apple.
If you look at the 30 apples and choose one of them, there is a ⅓ chance of it coming from the right, but if you first choose which trunk to climb (50%) and THEM pick an apple at random from that trunk, there is a ½ chance of it coming from the right.
If you pick the trunk you’re going to climb without looking at how many apples there are in each, the chances of it coming from any given trunk are the same as the chances of picking that trunk to begin with, no matter how many apples there are in each trunk. On the other hand, if you look at all the apples and decide which to pick and then use that to deduce the truck you have to climb, then that's the same as picking all the apples, putting them in a basket, choosing one at random and asking whether it came from the right or left. Those are two FUNDAMENTALLY different things, but it's hard to realize that in SBs version of the problem.
This is a cleverly devised Monty hall problem. At 4:24 it is correctly stated that she "doesn't learn any new information by being woken up" yet at 0:14 we are casually informed that she "knows all the details of the experiment". This is all the information she needs to be a thirder like me. Furthermore, Tom goes on to explain that it is a language ambiguity which as it happens is often the cause of paradoxes, and would be correct if she hadn't been given the additional information. It all goes to show just how carefully questions and suppositions need to be analysed and how easily influenced we can be by presentation. I love this video.
The Monty Hall problem starts with three equally likely probabilities, and then two of them are collapsed into one, leaving two unbalanced choices.
This is the opposite: it starts with two equally likely probabilities, then one of those two is split into two possibilities, which thirders erroneously set as each having the same likelihood as the other option.
The assumption made @7:00 in the video confuses me and I’m not sure it’s valid. Instead of making a different assumption about those conditional probabilities, I’m going to go through all the long math in this comment to explicitly calculate for the probabilities Tom assumes here are equal. With this, I will attempt to show how these two seemingly opposing views can be made more consistent.
Setup and Events:
Mon=It is Monday
Tue=It is Tuesday
H=Heads
T=Tails
Note that H and T are Complements and so are Mon and Tue.
Assuming fair coin:
P(H)=1/2
P(T)=1/2
Calculations:
P(Mon|T)=P(Mon AND T)/P(T)
=>1/2=P(Mon AND T)/(1/2)=2*P(Mon AND T)
==>P(Mon AND T)=1/4
P(Tue|T)=P(Tue AND T)/P(T)
=>1/2=P(Tue AND T)/(1/2)=2*P(Tue AND T)
==>P(Tue AND T)=1/4
==>P(Mon AND T)=P(Tue AND T)=1/4
P(Mon|H)=P(Mon AND H)/P(H)
=>1=P(Mon AND H)/(1/2)=2*P(Mon AND H)
==>P(Mon AND H)=1/2
P(Tue|H)=P(Tue AND H)/P(H)
=>0=P(Tue AND H)/(1/2)=2*P(Tue AND H)
==>P(Tue AND H)=0, this one is trivial
P(H|Mon)=P(Mon AND H)/P(Mon)
=>P(H|Mon)=(1/2)/P(Mon) (*)
P(T|Mon)=P(Mon AND T)/P(Mon)
=>P(T|Mon)=(1/4)/P(Mon)
By (*), P(T|Mon)=1-P(H|Mon)=1-(1/2)/P(Mon)=(P(Mon)-(1/2))/P(Mon)
=>P(T|Mon)=(P(Mon)-(1/2))/P(Mon)
=> (1/4)/P(Mon)= (P(Mon)-(1/2))/P(Mon)
=>1/4=P(Mon)-1/2
==>P(Mon)=1/4+1/2=3/4
==>P(Tue)=1-P(Mon)=1-3/4=1/4
==>P(Mon)=3/4 and P(Tue)=1/4
(*)==>P(H|Mon)=(1/2)/(3/4)=2/3
==>P(T|Mon)=1-P(H|Mon)=1-2/3=1/3
==>P(H|Mon)=2/3 and P(T|Mon)=1/3
P(H|Tue)=P(Tue AND H)/P(Tue)
=>0=0/(1/4)=0 trivially
P(T|Tue)=P(Tue AND T)/P(Tue)
1=P(Tue AND T)/P(Tue)=(1/4)/(1/4)=1, so all conditional probabilities in this setup are consistent.
Results:
P(Mon)=3/4
P(Tue)=1/4
P(Mon AND T)=P(Tue And T)=1/4
P(Mon AND H)=1/2
P(Tue AND H)=0
P(H|Mon)=2/3
P(T|Mon)=1/3
P(H|Tue)=0
P(T|Tue)=1
Therefore, we see that the assumption made in the video that P(T|Mon)=P(H|Mon) is false. While intuitively it’s true, we have calculated the probabilities to be 1/3 and 2/3 respectively, and thus knowing it’s Monday does clearly update your probability on what side of the coin was flipped. This is more like the “halfer” view in the sense that assuming nothing of the day, we still have 50% for the coin flip. However, we do get probabilities of 1/3 and 2/3 now, except here they represent the probability for Tails and Heads GIVEN you know it’s Monday so not completely the same as the “thirder” view. It’s still paradoxical to me though that knowing it’s Monday should affect the probability when both situations would have sleeping beauty inevitably being woken on Monday anyway. But this is just our intuition failing us, and the math tells us that Monday pushes the probability more in Heads’ favor. Please let me know if I made a mistake in any of my calculations.
Setup:
A=Now it is Day 1
B_i=Now it is day i for i in {2,3,4,…,N}
The B_i’s behave identically so we will use B to refer to any of them.
H=Heads
T=Tails
H*=T
A*=UB
Where * here is complements and U is union.
P(H)=1/2
P(T)=1/2
N=amount of times woken up if Tails is flipped
Calculations:
P(A|T)=P(A,T)/P(T)
1/N=P(A,T)/(1/2)=2*P(A,T)
==>P(A,T)=1/2N
P(B|T)=P(B,T)/P(T)
1/N=P(B,T)/(1/2)=2*P(B,T)
==>P(B,T)=1/2N
==>P(A,T)=P(B,T)=1/2N
P(A|H)=P(A,H)/P(H)
1=P(A,H)/(1/2)=2*P(A,H)
==>P(A,H)=1/2
P(B|H)=P(B,H)/P(H)
0=P(B,H)/(1/2)=2*P(B,H)
==>P(B,H)=0
P(H|A)=P(A,H)/P(A)
P(H|A)=(1/2)/P(A) (1)
P(T|A)=P(A,T)/P(A)
P(T|A)=(1/2N)/P(A)
P(T|A)=1-P(T*|A)=1-P(H|A)=1-(1/2)/P(A)=(P(A)-(1/2))/P(A)
==>P(A)>1/2
==> (1/2N)/P(A)= (P(A)-(1/2))/P(A)
==>1/2N=P(A)-1/2
==>P(A)=1/2N+1/2=1/2N+N/2N=(N+1)/2N
==>(N-1)P(B)=1-P(A)=1-(N+1)/2N=2N/2N-(N+1)/2N=(N-1)/2N
==>P(B)=1/2N
(1)==>P(H|A)=(1/2)/((N+1)/2N)=N/(N+1)
==>P(T|A)=1-P(H|A)=1-N/(N+1)=(N+1)/(N +1)-N/(N+1)=1/(N+1)
P(H|B)=P(B,H)/P(B)
0=0/P(B)
==>0=0
P(T|B)=P(B,T)/P(B)
1=P(B,T)/P(B)=(1/2N)/(1/2N)=1
Results:
P(A)=(N+1)/2N
P(B)=1/2N
P(A,T)=P(B,T)=1/2N
P(A,H)=1/2
P(B,H)=0
P(H|A)=N/(N+1)
P(T|A)=1/(N+1)
P(H|B)=0
P(T|B)=1
Case N=2:
P(A)=3/4
P(B)=1/4
P(A,T)=P(B,T)=1/4
P(A,H)=1/2
P(B,H)=0
P(H|A)=2/3*
P(T|A)=1/3*
P(H|B)=0
P(T|B)=1
Case N=1,000,000:
P(A)=1,000,001/2,000,000~1/2
P(B)=1/2,000,000~0
P(A,T)=P(B,T)=1/2,000,000~0
P(A,H)=1/2
P(B,H)=0
P(H|A)=1,000,000/1,00,001~1
P(T|A)=1/1,000,001~0
P(H|B)=0
P(T|B)=1
Case N=2(Original):
P(A)=3/4
P(B)=1/4
P(A,T)=P(B,T)=1/4
P(A,H)=1/2
P(B,H)=0
P(H|A)=2/3
P(T|A)=1/3
P(H|B)=0
P(T|B)=1
These calculations are all correct under the "halfer" point of view.
@@RaRa-eu9mw I defined the events and did the probability calculations. I didn’t assume anything of the situation under any “point of view.” How are these calculations then not consistent under the “thirder”point of view?
Where does the math go wrong?
@@Happy_Abe Sure. The issue is that the initial P(H)=1/2 is the probability that a fair coin lands on heads, but it then appears in the calculations as the probability that, when Sleeping Beauty is being asked the question, the coin landed on heads.
For the thirder point of view, P(H)=1/3 and so we get statements like P(Mon AND T)=1/3.
@@RaRa-eu9mw I’m not following you, we know it’s a fair coin so we start with the fact that P(H)=1/2
I’m not assuming this, we are given this information by the case of a fair coin. There’s no calculation I see(that doesn’t rely on further assumptions like made in the video that I addressed in the beginning of my comment) that would imply that P(H)=1/3. The probability that is actually 1/3 is the probability that Tails was flipped GIVEN it is now a Monday and there’s a 2/3 probability that Heads was flipped GIVEN it is now a Monday. Please provide the mathematical calculations showing that P(H)=1/3 without assuming that P(H|Mon) and P(T|Mon) are the same like in the video with no justification other than being intuitively true. While intuitively it makes sense to assume that, the calculations I did rely on no assumption and imply the contrary and are therefore more valid because they were derived, not assumed on intuition.
It's not that the tails is more likely. It's that she's more likely to be correct
and how is that not the same thing
Yep, those two things are answering fundamentally different questions. Which precludes both solutions being correct. Only one can be correct per the question that is asked
Isn't there a destinct difference between "What is the probability the coin was head?" and "Do you believe you are on a 'head'-branch of being woken up?"?
I mean the difference is "what is the probability the coin was head?" vs "what is the probability the coin was head given we wake you up?"
The second question is close to what Brady said with the bet of getting x value every time you are right, meaning the expected value is higher saying you are on the branch with more wakeups.
Also I don't think it can be healthy for sleeping beauty to get this medicine 999 times 😅😂
1:38 I somehow TOTALLY FORGOT that Brady has a kid! I feel like I want to congratulate him again, but it's been nearly two years since it happened 😂
One of the coolest things I ever saw was a math teacher prove that 1=2. He had to divide by zero to get there, which is not possible in math. I think thirders are "dividing by zero". A coin toss is always 50/50. In the same way, the effect can never "effect" the cause of the effect.
If you do get extra information, then the conditional probability of heads given that information could be different from 1/2. But in this case the point is that you do not get extra information, so I completely agree with 1/2.
I toss two coins and hide them. I tell you that the coins under my hands are not two heads. What is the probability that the first coin is a head?
I have a bomb wired to a fair 6 sided dice, so that if it rolls a 1, it will detonate, killing both of us. I roll the die and ask you what the roll was - what is the probability it was a 6 (same question for each other number)? What is the probability it was at least a 2?
By eliminating a result, *and knowing that result is eliminated*, you have changed the game and the conditional probabilities *based on the information you have*.
This just seems like an incomplete question, and Brady started to touch on that towards the end. The example he gave is "every correct answer gets you $1000" to which the answer would of course be tails, but the counter part is "if you were correct every time, then on Wednesday you get $1000" then your answer wouldn't matter as long as it was consistent (which the possibility of a changing answer is a separate rabbit hole). It reminds me of the video with Grant about choosing a "random" chord on a circle.
Is it just me or does the second assumption (coin flip given monday 50/50) just seem false? You can get the values without assuming this:
The distribution of the initial coin flip should be given at the start. If it isn't stated explicitly I assume it's 50/50, that is P(T)=P(H)=1/2. I understand the first assumption. Conditioned through the coin being Tails, we're looking at a particular experiment where you definitely woke up twice. So in that experiment there's no reason to believe the current day is more or less likely to be a monday. This would give is P(Mon|T)=P(Tue|T)=1/2. Okay, we don't need to assume anything else:
The probability P(Mon and T) is simply P(T)*P(Mon|T)=1/4. Same with P(Tue and T). However, P(Mon and H)=P(H)*P(Mon|H)=1/2*1=1/2. That means without assuming anything else we get back probability of heads = P(H)=1/2 (as we should). Assuming P(H|Mon)=P(T|Mon) is creating a contradiction with their real values:
P(H|Mon)=P(H and Mon)/(P(H and Mon)+P(T and Mon))=2/3, and P(T|Mon)=1/3.
To me there's no paradox, the other value arises from an assumption that results in a contradiction.
i have alwaays been in admiration of the acting skills of the guy behind the camera asking questions
I feel there's some sort of unsettling implication to this problem: Are we usually the Sleeping Beauty, trying to find the truthful probability of an event from a biased position?
We could do perfect math, but still the answer could be wrong because we can only look at things from our perspective.
That's a great Insight. The sleeping beauty problem was actually invented by Arnold Zuboff and named by Adam Elga. In Zuboff's Work, he talks about the perspectival Nature of probability. This is important to the solution of the sleeping beauty problem.
You are totally right. The universe doesn't pick if you win on a lottery ticket when you scratch it.
It is determined when it is printed, delivered and stored. By the time you purchase the one on top, it's no longer probability unless YOU ignore the content of the pile.
You don't know the outcome yet, but that outcome now has a 100% chance of occuring, assuming you scratch that ticket at some point in the future.
@@laplongejunior Me or them?
@@tedyplay4745
Both? Everybody.
I reckon the thirder aproach is the correct one. Yes, in a vaccum, the coin is fair, so 50%. But, with the rules of the game, the sheer fact that she is woken up is information itself.
Imagine, instead of being woken 1 or 2 times, we make it so if the coin was heads, she is woken once; if tails, not at all.
If she is then woken up, she knows for a fact the coin was heads, or she wouldn't be awake on the first place. The original situation is subtler, but the fact that she is woken twice on a head makes it so there are more opportunities for her answer heads. 2 out of 3 times she is awoken, the coin is heads, since she can't know which of the three posibilities it is.
It is the rules of the game which affect the probability, not the fairness of the coin.
I feel like this is kinda similar to the observer effect in quantum mechanics. The simple fact that the question is asked to someone involved in the problem and thus "inside" of the probability tree itself, changes the meaning of the question and therefore its answer. Like if the probability itself was observer-dependent.
Exactly!
Rather than extend the experiement to 100 days, imagine instead that the experiment only lasts 1 day. So, if the coin flip is tails, sleeping beauty is woken up 1 time, but if the coin slip is heads, she is woken up 0 times. Then of course, if she is awake, she knows with complete certainty that the flip was tails!! But the "halfer" logic would says it's still only a 50% chance of tails! Point is, the fact of being awake does present new information, while the "halfer" argument depends on a total absense of new information.
Going back to the original experiement, P(Heads)=P(Heads|Monday)*P(Monday)+P(Heads|Tuesday)*P(Tuesday) = 0.5*P(Monday)+0*P(Tuesday), so P(Heads)
I remember posting this one to Metafilter back in 2004 or so. The comments section eventually came to the conclusion that there is no paradox, it's just a badly defined question that we're asking Sleeping Beauty.
Completely agree. I had a rant about it in the comment section on Veritasium's video about it back in February. If it was voice acted, the voice would be laud and angry 🙂.
Quoting myself:
"Basically - the only problem I can see, is the problem that whoever came up with this *really* can't ask questions. The discussion about 50% vs 1/3 is more about *what the question is* than anything else.
There's a reason I *kinda* prefer computer science over g.d. philosophy.
" - Me, Feb 2023.
I was... not happy.
Currently the vote on Veritasium's video is 193,340 "Thirders", 85,660 "Halfers".
@@phizc I would expect all mathematicians sufficiently trained in probability theory to agree that the question asked is P(heads | you woke up), and that, since you always wake up, P(heads | wake-up) = P(heads AND wake-up) / P(wake-up) = P(heads) / 1 = 1/2.
It is interesting that this simple exercise in conditional probabilities has become an accepted philosophical problem that has attracted dozens of publicatons, many of which are full of quite silly statements. This shows that it is best to ignore all the hand-wavy "probability" arguments by philosophers about whether we live in a simulation and similar questions.
It's actually very clearly defined, the question we are asking is P(heads | awake-now) - absolutely no philosopher or mathematician who has ever written on this problem disagrees.
@@dominiks5068 except that the philosophers are typically unable to calculate this simple conditional probability 😃. I agree with you that it is the correct formulation of what is asked.
Interesting problem, but I believe I agree with most people here that there isn't really a paradox. The correct probability is 1/2. The entire "thirder" argument is geared towards answering the different question "what is the probably that you are awake now AND the coin was heads."
The betting argument is also good, where it's clear the thirder strategy is the better one but only because when the coin is T you get to "play" more often.
Consider a modification: since Sleeping Beauty never gets any new information, we may assume that she will give the same answer every time. So since all the wake-ups under tails are dependent, they are all combined into one event (they all get the same answer and either all happen or none do). Now the 50-50 answer is pretty clear.
So I think that's my final answer: the probably is 50-50 and the other strategies are all answering (or strategizing for) different questions.
Cool problem, but not a true paradox in the sense of something self contradictory (I've never liked the other types also getting called "paradoxes" but that's not up to me).
Can you really count 3 possibilities, when 2 of them are perfectly dependent on each other?
It's not only about the question but about how her answer is evaluated. If you want to count how often she is right over all wake ups, she should say tails (correct chance is 2/3). If you count how often she was right in all wake ups after a toss she can say either (chance is 1/2). So the question is: Is one run off the experiment considered one wake up or one toss?
The simple answer that I came to with this is that you asking the question does not retroactively change the probability of the coin flip. Even in the 999 scenario, the probability of a coin flip is 50/50, no matter what. I'm with Brady on this one.
Perception does not alter the statistics, and if no new information is being conveyed (which is the point of the amnesia) then there is no way to know whether it was a heads or a tails. It's 50/50 regardless. Funnily enough, this is like the opposite of the Monty Hall problem but with a similar solution. People are perceiving 2 doors and assuming the chance is 50/50, when in fact the probability was decided when there were 3 doors, so the odds are in thirds. This is the opposite. The perception is that there are 3 situations you could be in, but in reality there are only 2.
I feel like most of these mathematical "paradoxes" are the result of ambiguity in the question being asked. Even here, it was "well, if you ask it this way, it's 50/50 but if you ask it this other way then it's 1 in 3...or 1 in 1000".
This is similar to The Monty Hall Problem problem, which is that there are two questions, which can be expressed in very similar ways - subtle details distinguishing one from the other - and they can easily be confused.
Once one question is asked, though, it has one answer. The listener might be expecting a different question, but if they correctly answer a question that's materially different from the one asked, they have not correctly answered the question asked. And the asker might have in mind one question, but stumble on that subtlety when putting it into words, and so the asked question can be different from an intended question. And it can even happen that both inversions happen, the intended question gets its correct answer via the asking of a different question.
But once one question is asked, it has one answer.
If sleeping beauty is asked to make a wager which will be evaluated once on Wednesday (and if it was tails and she answered differently the different times she was asked, her most recent wager is the one which is used (alternatively, if tails, she is only asked on one of the two days)) then it makes sense for her to use 50/50 .
So, if what wagers will be available and how they will be evaluated, is known beforehand, there are no paradoxes in “how should she bet”. The only difficulty is the question “what types of wager odds should we consider to define the probability?”.
Here’s a variant:
She is told beforehand that an amount of money X has been chosen, but she won’t be told the value of X until Wednesday.
If heads, she will be offered a wager for an amount which is of size 2X, but she won’t be told the amount (as this would give information about whether heads or tails), and if tails, she will on each day be offered a separate wager of size X (but again, will not be told that it is of size X rather than 2X).
(The wagers are each about which way the coin landed)
In this case, the wager odds she should accept would be those in line with a 50/50 chance.
From the half-ist perspective, the weightings on both sides are equal, so the way-to-bet is unchanged from the true 50/50
and from the third-ist perspective, the weighting added to the heads case shifts how one should bet away from the 1/3 probability to become 50/50 .
Another variant:
Instead of whether she wakes up twice, the coin only determines whether she will be offered on Tuesday the option to bet. (If heads, on Tuesday she will not be offered the wager.)
In this case, it seems that there are 4 equally likely scenarios, {heads,tails}x{Monday,Tuesday}
But, on the days she is given the option to bet, the information she has available based on which to make the bet, is identical to the information she has in the normal version of the scenario, so it sorta seems like the probabilities should be the same.
In this case, P(Heads|monday and offered a bet )=P(tails|monday and offered a bet)
But P(Heads|Tuesday and offered a bet) = 0, and P(tails|Tuesday and offered a bet) = 1
P(Monday) = P(Tuesday), but P(Monday|offered a bet) = (2/4)/(3/4)=(2/3)
And P(tails|offered a bet) = (2/4)/(3/4)=(2/3)
So, in a sense, if “am awake” in the original formulation is substituted for “am offered a bet” in this variant,
it kind of seems like the third-ist position is essentially conditioning on “I am awake” as evidence about what day it is?
Hm.
I’m undecided as to whether this is “an actual question which has an answer”, or whether it is a question of semantics of “what do we want these words to mean” where the actual “how-things are” is clear.
There is no paradox whatsoever. The correct answer is 0.5, since there is only one independent event in the experiment - the coin toss. Therefore the probablility of waking up with heads and tail is always 1/2. The probability of waking up on Monday with Tails is 1/4 probability of waking up on Tuesday with Tails is 1/4 and probability of waking up on Monday with Heads is 1/2.
I'd say the halfer answers the question "What is the probability of a correct answer if we asked if the coin landed on heads (or tails)"
While the thirder answers the question: "What is the expected number of correct answers if we asked if the coin landed on heads (or tails)".
These are not the same question, hence different answers.
From the point of view of SB after being asked the question, they have the same answer.
I was a thirder, but now I'm a halfer. But I think the way if interpreting it is as saying heads and getting it right gives you 1 valid answer and saying getting it wrong gives you 2 wrong answers. So it's a 50/50, but the expected value of right answers is 1/3.
yeah!!! i enthusiastically agree
I'm fascinated by the distinction in Brady's head, between "what is the probability that an event happened" and "what is the probability that we're on the branch where the event happened". To me they seem like the same thing.
I feel like he's conflating the first one with "before the event happened, what _was_ the probability that the event was going to happen?". But that's different from asking "what _is_ the probability that it _did_ happen?"
The literal answer to the second one is 100% (if it happened) or 0% (if it didn't). But what we usually mean is "what level of confidence should you assign to it, _given the information you have available_ ?"
Ah, the confusion between the question she was asked, "What is the probability the coin came up heads?" and the question she was *not* asked "What is the probability that a coin flip will come up heads?"
The first asks for the conditional probability given that you woke up. Since you wake up always, the answer is still 1/2.
P(heads | wake up) = P(heads & wake up) / P(wake up) = P(heads) / 1 = 1/2.
@@ronald3836 Ah, but since you wake up always the answer is heads 1/3.
@@reidflemingworldstoughestm1394 no, the conditional probability is calculated as I indicated. The outcome is 1/2. Translated into words: since you know beforehand that you will wake up, you don't gain any information from waking up. So the conditional probability of heads given that you woke up is equal to the probability of heads, which is 1/2 (fair coin).
@@ronald3836 No. That is the part you got wrong. Gained knowledge is a red herring. The best answer is heads 1/3.
@@reidflemingworldstoughestm1394 I like the way that Ronald explains his answer with reasoning and equations, and you just assert your answer with nothing but bluff and bluster. I guess the two of you are equally likely to be right! ;)
"What is a probability that the coin flip was heads?" is a very different question than "What is the probability that when you say the coin flip was heads, you are correct?" In the first case, the answer is 1/2 and in the second one, the answer is 1/3.
Is there a possibility one of these statements is true without the other? If not, they are the same probability.
How is "A is true" different from "if you say A is true, you're correct", those are the same thing.
I'm still staying with the 50% 50% answer.The explanation using probabilities arithmetics just tells you that the 3 scenarios are equally likely to occur, but that doesn't mean that they are separate probabilities, so tthey shouldn't be treated like they can occur independently, it can't be Tuesday if it wasn't Monday first, they're both linked, so yeah, their odds are the same, because they're literally the same event, just at a different time.