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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.

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.

Numberphile exposure has turned Brady into a bona fide mathematician, and I am firmly here for it.

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?

Can honestly say I never expected to be cast in the role of sleeping beauty in a Numberphile video…

I love when a thought experiment is so strange that you also have to imagine that consent was given

Have you also felt this discomfort when the needle goes into her head?

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)

Brady killed it in this one! It was a very perspicacious way to avoid the paradox.

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.

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.

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 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.

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.

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 agree its not a paradox, theres a semantic switch up to conflate probabilities for two different things

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.

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.

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 Brady really cuts to the heart of this paradox, the specific wording of the question changes the answer. Any ambiguity in the question is where the "paradox" arises. And Tom Crawford is a gem, his passion, enthusiasm and knowledge of mathematics is something special!

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.

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)

This isn't a paradox, it's a semantic issue and a limitation of the English language. Clarifying the question would inevitably lead to one of the two answers. This isn't a math problem, but a language one.

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.

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.

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.

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.

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.

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.

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.

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!

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.

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.

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.

It's not that the tails is more likely. It's that she's more likely to be correct

Absolutely love Tom's enthusiasm, all the videos featuring him are a treat (even for a topic as infuriating as that one!)

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.

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.

Are you asking sleeping beauty: (1) "did we flip a head or a tail", (2) "what is the chance we flipped a head or tail", or (3) "what is the chance of a heads, given that we have woken up"

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.

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.

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

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 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 genuinely love the visceral GLEE on Tom’s face through this.

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.

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.

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.

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.

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.

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.

As with most of these probability paradoxes, it all boils down to understanding/specifying further exactly what the question being asked is

Hey! Brady's a Dad! Congratulations

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%.

This strange and ambiguous problem is exactly what happens when mathematicians have too much free time on their hands.

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.

I tried to work it out with conditional probability, etc. and got all turned around. For me the more intuitive approach was to ask, if you did this experiment 100 times, how many total wake-ups would there be and how many wake-ups with heads would there be? Answer: 150 and 50, so 50/150 = 1/3

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.

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.

You can simplify this question by creating this experiment: box 1 has one ball, identical box 2 has two balls. You pick a random box with a 50/50 chance and draw a ball. What is the probability that the first ball was chosen?

The problem here is that the Monday and Tuesday for Tails are treated as independent of eachother when they are not. P(T | Tue) = 1, since the only situation where Tuesday is an option is when the coin was Tails. The only way the thirder philosophy makes sense is if you conduct this experiment multiple times, record all of the answers and then mix up the responses into one sample - then approximately 2/3 of the total responses will be on the 'tails path'.

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.

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

Did you know, a poll was conducted of people's opinions. 50% were halfers, but 2/3 were thirders.

Can you really count 3 possibilities, when 2 of them are perfectly dependent on each other?

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.

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.

This feels like it could rdebunk the hypothesis we're living in a simulation.

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_ ?"

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!

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.

The trouble with alternate timelines isn’t the math, it’s the tenses: “That coin, which will have had a 50% chance of being heads, has a 66% chance of having been tails.”

Modification: Split a deck of cards into two piles: 51 and 1. Flip the coin. If heads, wake her up 51 times and show the next card in the deck each time. If tales, wake her up 51 times and show her the same card each time. Each day, ask her if she thinks the coin was head or tails.

I love that they drew him as the sleeping beauty hhaahhaha

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.

Is there a "Quarter-er" theory? There are only 4 possible situations: Monday + Tails + Awake Monday + Heads + Awake Tuesday + Tails + Awake Tuesday + Heads + Asleep You state that Sleeping Beauty has no new information; however she *does* know that she is awake - which *IS* new information. Given that you are awake, Tuesday + Heads + Asleep is eliminated as a possibility. This is literally the Monty Hall problem, and the fact that you are awake is the same as having opened door number 3, which means that one of the options has been removed. In other words: 4 Doors, 2 prizes. Picking a door at this point is the same as flipping a coin. You don't know if you're correct or not. The host opening an empty door is the same as waking you up. There is now one less "bad" choice, so the odds of the other two being 'correct' is now 2/3rds instead of 1/2. So, if the answer to the Monty Hall problem is "You should always switch"; then *as Sleeping Beauty* you should always guess that the coin came up Tails. I feel like the more intellectually honest way to ask this question is "What are the odds that today is Tuesday?"

This is exactly the same internal tension I feel about the antropic principle.

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 😅😂

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?"

"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.

The problem with the conditional probability math is that you are assuming that Monday(Tails) and Tuesday are separate events. Given the construction of the question, they are the same event. Then, effectively, your reasoning follows: a=a=b, and these are the only options, so by total probability a=a=b=1/3. This is wrong - if a is the same event as a, then total probability would give you a=a=b=1/2. If you had asked a different question, then Monday(Tails) and Tuesday may be separate events, but you didn't.

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.

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.

Much less of a paradox than it seems. It's 50% / 50%, always. It only becomes confusing once you begin to account for "being right about the outcome of one particular flip", which is kind of irrelevant question. You are not being asked to be right about a particular flip (or rewarded for answering correctly), you are asked about probabilities of an overall flipping outcomes.

Regarding the point that if you got money deposited in your account every time you picked correctly so you should pick tails, then consider this alternative. Suppose Sleeping Beauty gets $1m if she correctly picks heads, $0 if she is incorrect, and gets $100 for every tails she responds with a potential of getting asked that 100 times. Would you still pick tails?

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.

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 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 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.

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.

I saw the thumbnail and immediately knew who it was.

Keep it simple: if I flip a coin and tell you what it landed on, there is still a 50/50 that the coin had landed heads or tail (the half position) but you have 100% accuracy in guessing the answer (analogous to the third position). It just depends what you interpret the question to be asking

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.

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.

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?

Say instead of a coin, they roll a six-sided die. If they roll a 1-5, you only get woken up on the Monday, but if they roll a 6, you get woken up every day from Monday to Friday. You wake up, which scenario do you think you’re in?

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 don't understand, the halfer approach doesn't even stand the test of... well... test? If you conducted the experiment 1000 times, you would wake her up 1500 times, of which 1000 would be as a result of Tails and only 500 as a result of Heads. Why is 50% a valid answer for her?