## [0862] The Sleeping Yuna Problem

└ posted on Monday, 20 February 2017, by Novil

Each of those little panels with Yuna in them is drawn individually. That’s dedication to the art form that only nutcases like Bill Watterson would appreciate.

**Ye Thuza:** Rise and shine, Yuna! Because you were such an insufferable brat last week, I decided to carry out a devilish experiment on you.
**Yuna:** Really?! But I don’t have any tentacle arms!
**Ye Thuza:** It’s more of a mind game.

**Ye Thuza:** In this experiment, the mad scientist – that’s me – puts the human guinea pig – that’s you – to sleep on Sunday. Once or twice, the test subject will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget the awakening. A coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, the test subject will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday *and* Tuesday. In any case, the experiment will end on Wednesday without any further interview.

**Ye Thuza:** This means, it’s either Monday or Tuesday today. Now I ask you: what is your credence now for the proposition that the coin landed heads?gezeigt hat?
**Yuna:** That’s easy! It’s 50%!
**Ye Thuza:** Oh, is that so?

**Yuna:** No, wait…! It’s a third! … Hmm…
**Ye Thuza:** Even if I say that the coin is still lying on the kitchen table?

That’s easy if you have a protocol against mind attacks and time manipulation. Say you’ll scratch your arm in case you are confronted with the possibility of your memory getting compromised. Then, you wake up and get asked that. You check your arm, and if it isn’t scratched, you can tell it’s a 50% chance. If your arm is scratched and you don’t remember doing that, you know it’s 0% and it’s Tuesday.

Of all people in this comic, Yuna would be the most likely to have some protocol like that 😉

Ha! You just want to troll us into creating a flame war while attempting to answer the riddle. I see right through you … and cannot resist anyway.

There are more interviews in the “Tails” timeline than in the “Heads” timeline, so the answer to the question “

what is the probability of ‘Heads’ being the correct answer in any of the three interviews” would be one third.But that is

notthe question. What Ye Thuza aks for is thebelief, i.e., Yuna’scurrentstate of mind, which is the same in all three interviews.Doing multiple interviews does not change the state of the coin; the probability of being in the “Heads” timeline is exactly the same as being in the “Tails” timeline, regardless of the number of interviews.

1. Why would she not know if it’s Monday or Tuesday? Are you saying that amnesia-inducing drug makes her sleep through a whole day?! She’ll be awfully dehydrated.

2. Why would it be one third? Which of the three possible awakenings she’s in is clearly not drawn from a uniform distribution, if that’s what we’re supposed to think. The heads/Monday case has 50% prior probability, while both the tails/Monday and tails/Tuesday cases each have 25%. The given information doesn’t change anything (since we aren’t told what day it is), so nothing has changed from the priors. It’s still 50%.

Well, the problem with stochastics is, it can be a bit confusing. It’s quite easy to make a mistake there if it gets a bit more complex, even for a genius like Yuna.

And yes, it is 50%, at least in theory. What happens in these two cases doesn’t influence that probability.

On the other hand, the real probability does depend on the coin, it will never be exactly 50% in real life.

50%

@ Vidad:I like your logic.

Vidadwrote:I wonder if this will go live to the general public on Tuesday just to mess with us.

@ r4m0n:There’s no need for that, this strip has already messed with some people here.

Hm 100% for Head. In a normal situation yes 50% in this special case 100. I don’t think Ye Thuza would let pass her a compleat day in school 😛

Monty Hall would be proud. Good job Novil. 😉

Cervisiawrote:Actually, there is no proper answer since any answer given by Yuna would be correct, and remain correct despite changing during interview.

Experiment purpose (one of possible puposes, anyway) would be to determine how Yuna deals with “time/memory” situation rather than testing her ability to solve probability question.

But to deal with “what day is it” question – assuming that kitchen table is something they actually use to eat food in this household, and stray coins would not survive there for long, it should still be Monday.

The interesting thing about this problem is that its from Yuna’s point of view.

She’s not asked what the chances of an arbitrary coin flip being heads, shes asked what the probabilities are that the coin has ALREADY landed on heads.

And since she would only get interviewed once if it was heads and twice if it was tails, in her position its more likely that the coin did not land on heads, thus the “one third” answer and subsequent second-guessing

It’s monday. This isn’t Gaia.

It is Sunday where I am!

*mind blown*

@ r4m0n:Now you made me look all the timestamps on the comments and it looks like a weird timelords conversation

@ Eydel:But.. what… how.. ???

Have the timestamps been tampered with? How is it possible that people write comments over a week before the comic is posted?

@ Senjiu:Patreon patrons get access to some of the comics earlier, but the time-stamp for the comic remains for the day everybody gets it.

Although I do like the idea we’re time-lords.

I’m not even going to try. Real life makes me feel stupid enough as is.

Vidadwrote:I’ll add that to my CV.

That’s a harsh but necessary lesson Ye thuza is teaching her daughter here. A proud young prodigy like Yuna will have nightmares upon not being able to solve properly that riddle, talk about how to learn humility. 🙂

@ qq:I propose that is irrelevant, especially since that piece of information has specifically been noted and mentioned. It could specifically have been left there from Monday on to Tuesday, with the knowledge that it would make the subject doubt herself. (Do keep in mind that the entire experiment was prompted because Yuna was “a brat” this last week, and a probable motivation is psychological punishment. Indeed, there may have not been a coin flip at all, nor any amnesiac drugs.)

you demon! This nefarious plot!!

@ Senjiu:Damn Patreon patrons; they mean well, but alter time and space on a weekly basis.

Which door to choose! What is the chance of winning the big prize?!? ;P

Ok, if there’s one and only one toss coin…

But let’s have other elements. First, it’s Early monday morning. Why? a kitchen table is very used, and hardly the mad scientist would keep it there after breakfast, lunch, kids homework, dinner, adults dealing with papers.

So, it is the first awakening. The question is if she will be awakened once more? .

I would guess the coin landed heads, She will not be questioned more, because Ye Thusa is going all-in on this monday questioning.

So,my answer: 100% heads

I’ve considered a few methods of calculation, which yield different results. However, the only correct way to calculate probabilities like this is from the past the the present, because the simple lack of information does not imply even distribution.

Now the first action is, interestingly, Ye Thuza deciding whether to flip the coin before or after the Monday interview. In the absence of additional information, let’s set this at 50%. So we have 50% non-flipped Monday, 50% flipped Monday.

In the case of flipped Monday, we can now split to 25% Monday heads and 25% Monday tails. Monday heads ends the experiment, so that probability is final.

Next step, if we’re on Monday tails, we can reach Tuesday tails. Now, it might be tempting to handwave and say we have a 50% chance of having already advanced, but that would be a huge hand-wave because probabilities don’t really work on the timescale like that. Fortunately, it doesn’t matter, since the odds of being on heads are 0% on both cases.

Now, what about non-flipped Monday? Since flipping heads after the interview would not result in an additional interview, there is 0% chance of getting heads in that case. And since there’s a 0% chance of having heads during said interview, we once again don’t have to hand-wave over time, and instead conclude that this entire subtree yields 0% chance of a heads interview (You’d either have a non-flipped interview, or a tails interview, at indeterminable distributions).

So, taking it all together, the odds are 25%.

Alternately, if the odds of Ye Thuza flipping the first coin before the interview is known to be x%, instead of being 50%, then the odds of a heads interview are x/2.

Whew, that was a mind-bender.

Took me a while to realize that it was Monday/Tuesday of the same night so that Yuna can’t just remember whether Monday daytime happened. XD

100% if it’s Monday, 0% if it’s Tuesday and we can’t remember what happened Monday.

Without knowledge of whether it’s Monday or Tuesday the best answer we can give is that there was a 50% chance of the coin landing on heads.

I’ve been avidly reading Sandra and Woo since late 2010.

I’ve never made a comment until now since I feel compelled by no one defending what I think to be right

My answer would be 1/3 because initially when the coin is flipped, there is a 1/2 chance of it being heads.

However, waking up for the interview adds a change in variables.

In any one point of waking up, there is a 2/3 possibility that you have been woken up due to tails. But 1/3 due to heads.

So it’s 1/3.

If Novil is reading this, I’d like to thank you for all the joy this comic has given me since I was 14.

It fits my humor perfectly, I love the nerdiness, art, political views, steganography, how well informed and thought out every comic is even despite readers possibly missing out on detail- and so much more.

Like a good coffee which spoils my taste to anything less, this comic has numbed me to many other reads just because they’re never as piquant a blend. So I read Gaia.

Sandra and woo is actually my browser homepage.

I am quite disappointed the cursed adventure hasn’t come out though.

@ Shorts:That is exactly true. In a different example which may illustrate the problem in a more intuitive way: imagine I tell yuna that if the coin was heads, she will be transferred into a red room 1 upstairs with no windows, and if the coin was tails, she will be transferred into a red room 2 or 3 downstairs with no windows. Since there are no windows, she cannot tell whether she is upstairs or downstairs or whether it is room 1, 2 or 3. Hence the variable “upstairs or downstairs” or “being in room 1, 2 or 3” does not influence the probability, as it is not determinable, and therefore provides no information. So if she finds herself in a red room, the coin has been tossed, but there is no more information that is given. Hence the probability is only determined by the coin toss. Similarly, she cannot tell whether it is monday or tuesday, or whether she was given a potion or not, and therefore such information irrelevant. Since she has woken up, the coin was tossed, hence it is of equal probability that the outcome was H or T.

It’s 50%

She’s always going to experience being woken up. Since she doesn’t remember the first interview, the second interview is irrelevant.

The fact that there is 2 interviews doesn’t change the odds of it being a 1 in 2 chance.

There is a similar problem involving gold and silver coins in 2 boxes. One contains a single gold and a single silver, and the other contains 2 golds. In this scenario, you are pull a gold coin from a box. You are then asked what the probability of having drawn from the double or single gold boxes.

The answer is a 1/3 chance for the single gold box and a 2/3 chance for the double gold. One may think it’s because there is 3 gold coins and 2 out of 3 of them are in the double gold, but the reality is that the number of coins is irrelevant. It’s about the ratio of coins for each box. If the double gold box had just 1 gold coin and nothing else, then the answer would still be the same, since you have 100% chance of drawing a gold from that box, and a 50% chance from the other. a 1:2 ration out of 3 total scenarios yields the same results: 1/3 and 2/3.

So in this problem you have:

a 100% chance to be interviewed and asked the question if it was heads

or a 100% chance to be interviewed and asked the question if it was tails

The ratio of probabilities is 1:1 out of 2 total scenarios and thus means both possibilities are 50%.

A lot of people seem to be under the assumption that because there are three interviews they are all equally likely. This is incorrect. There is a 50% chance of it being Monday and the coin having flipped heads (since the chance that the interview is on Monday given that the coin came up heads is 100%), there is a 25% chance that it is a Monday and the coin came up tails, and there is a 25% chance that is is Tuesday and the coin came up tails.

These are the raw probabilities – with no information these are the possible outcomes. So now we use the information we have to see if it affects these probabilities.

However the only information we have is that she is in one of the interviews (i’m not counting the coin being on the table as information since there are many interpretations of what that could mean). This doesn’t get rid of any possibilities, so the probabilities are unaffected.

So it’s still 50/50 whether the coin is heads or tails.

@ Radircs:You missed something; even if it is monday we could still be on the experiment track that’ll lead to the interview tuesday, which leads to a 50% probability, as what is questioned is the probability of the coin toss.

Rise and shine, Mister Freeman. Rise and… shine.

@ Senjiu:Maybe this is the interview and we’ve slept through a whole week. Lower the dosage, Ye Thuza…

The “obvious” answer is 50%, There is a problem with that I will mention but I think the obvious answer is also the right answer here.

Lets forget for the moment about “Bayes’ theorem” and make the problem simple.

Heads is 50% by definition and Tails is 50%.

If Heads then it is Monday, If Tails then its 50% Monday and 50% Tuesday.

Summing up the probabilities:

50% Heads-Monday

25% Tails-Monday

25% Tails-Tuesday

So the initial reactions of Yuna seems right. But there is a problem. The question is about YOUR (Yuna) CREDENCE. This certainly baffles Yuna but I still think the above probabilities are right. I do not see a reason to make equal so the right answer would be 1/3 (33.3%)

I’m firmly a “1/3” on this one. Note, most importantly, that they asked about her personal credence, not any sort of “objective” probability.

Imagine running the experiment many times. If she answers heads every time, she will be right 1/3 of the time. If she answers tails every time, she will be right 2/3 of the time. In particular, in 1/3 of these scenarios (in the long run, which is how we are actually supposed to measure probabilities in principle), she will have “heads” be the correct answer, so the correct credence to “heads” is 1/3.

And the Wikipedia link for those who want such things, but don’t know the name of the problem: https://en.wikipedia.org/wiki/Sleeping_Beauty_problem

Someone’s been playing

Zero Escape, I see.The scenario is turning the Monty Hall problem — which Yuna DEFINITELY knows about — on its head with a bunch of wibbly-wobbly timey-wimey stuff. Although there is a surface similarity to Monty Hall, here the intuitive answer is correct: despite any WHICH INTERVIEW IS THIS nonsense, the coin flip happened once. No conditional probabilities or new information exist. The H probability (50%) is trivial.

Yuna’s mom is obfuscating a problem that sounds like Monty Hall in order to mess with her (and first thing in the morning, no less!).

@ Magnema:There is a problem though. If the experiment is every week it is true that only 1/3 of the times asked will be Heads but why would she weighs these the same? I say in the long run 50% it will be Heads-Monday and 50% it will be Tails-Tuesday, since she will have amnesia for Tails-Monday. I would not be that firmly for the “right” answer…

@ nescience:Ahhh I had never heard of Sleeping Beauty! So yeah, the answer is definitely “mom, you’re messing with me.”

Let’s assume that the coin landed on tail. In this case Yuna will be interviewed twice once on monday and once on tuesday. due to the amnesia potion she can not tell the two interviews apart. This means that in that situation she have a 1/2 chance of this being monday and 1/2 for it being tuesday.

Let’s assume the coin landed on head. Now the chances are 1 for monday and 0 for tuesday.

Now let’s not assume the result of the coin toss. In this case we have 1/2 chance for head and 1/2 for tail. We get that the overall chance for monday are: 1/2*1 + 1/2*1/2 = 3/4. and for tuesday: 1/2*0 + 1/2*1/2 = 1/4.

There is no monty hall effect or any other such surprise because the amnesia potion guarantees information loss between guesses. Therefore the guess will be equal in both monday and tuesday.

gezeigt hat? In the script below the comic.

@ r4m0n:The real solution is to stare back and slump in the bed while saying “The answer doesn’t matter. My mother is willing to drug me for her own amusement.” Then lay back down, roll over and say nothing else for the rest of the day. Yuna blew that chance when she made her 50% guess.

@ r4m0n:I’ll go with 1/3.

If the experiment is repeated an infinite amount of time, Yuna will wake up a third of the time on heads, and two thirds on tails.

“But I don’t have any tentacle arms” – Was that a reference to assasination classroom? xD

MrSanchezwrote:Yeah, that one needs explaining to me. So very funny out of context anyway though!

Alright, time to do some mathematics.

Walking into this there’s a 50% chance for heads and tails.

If the coin landed heads, there’s a 100% chance of this being Monday.

If the coin landed tails, there’s a 50/50 chance of it being monday/tuesday.

That means a total of a 75% chance of it being monday.

And therefore a 25% chance of tuesday.

On a tuesday, there’s therefore a 100% chance of the coin landing tails.

On a monday, there’s a 50/50 chance of the coin landing on either side.

That means the odds of the coin landing heads is

the odds of it being monday * the odds of heads on a monday = 0.75 * 0.5 = 0.375 = 37.5%

This matches the odds of the coin landing tails:

The odds of being monday * the odds of tails on a monday + the odds of tuesday = 0.75 * 0.5 + 0.25 = 0.375 + 0.25 = 0.625 = 62.5%

0.375 + 0.625 = 1

So there’s a 37.5% chance of the coin being heads, i.e. slightly more than one third.

Don’t take this at face value though, I’m not a math expert.

I sometimes think that the only way to properly answer questions like this is to know which question is *expected* by the person who does the asking.

Otherwise, as illustrated here, everyone has a theory and can back it up convincingly enough, whereas the *real* answer is already known in advance – which answer heavily influenced the exposition and phrasing of the question. There aren’t enough facts – there’s a surplus of conjecture, not the least of which is the difference between “How many times did the coin land on heads?” and “How likely do you think it is that it landed on heads, considering this may be your first-and-only-awakening, first-awakening-out-of-two or second-and-last-awakening?”. It’s not a math/statistics problem, it’s a riddle.

And man, I hate riddles. 😛 Even if you come up with a right answer, you’re wrong unless you’ve come up with the answer the riddler intended.

Let’s say we do the experiment 100 times.

Statistically, 50 times the coin will show heads, 50 times is shows tails.

This means there are 50 interviews on Monday where the coin showed heads, 50 interviews on Monday where the coin showed tails and 50 interviews on Tuesday where the coin showed tails.

So in total, there are 150 interviews during our experiment. On 100 of them, the coin showed tails, on 50 the coin showed heads.

So for any randomly selected interview, the probability that the coin showed heads is 50/150 = 1/3.

The answer is therefore 1/3.

@Invenblocker: I just came to the same result on the german comments.