## [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?

Uh, okay…here’s my solution

Ultimately, the chances of a coin-flip are 50/50. So the chances of it being monday with heads is 50%, and Monday or tuesday with Tails also 50%. Since you can’t determine the day on the taisl route, the probability of it being Tuesday/Tails are 25%, and Monday/Tails 25%. So we have 75% Monday, 25% Tuesday.

If it’s Tuesday, it’s tails. So let’s put down 25% for Tails as a given. But if it’s Monday…well, that’s the pickle. If you’d know it’s monday, chances for the experiment are still 50/50, but it’s only a 75% for that, so…37.5/37.5. Meaning, Heads are at 37.5% and 62.5% Tails.

But that’s me just waking up^^. I could still explain the difference by Tails being asked on two different days instead of one, therefor being asked about it has a slightly higher possibility…

Uh… You guys realise that the coin is only flipped ONCE, right? Depending on the result of that flip, it’s decided if there are one interview or two, but the flip only happens once.

I don’t even see how it can NOT be 50%.

Oooh, that’s a very mean thing to do with a math genius. The first version of the question would at least still be solvable, although Ye Thuza already traps her by giving inaccurate Information, she does not state whether the coin flip on Monday has to be made before or after the interview, so Yuna does not know if she has to take the possibility into account that it is monday and the coin hasn’t yet landed on any side at all. (If we assume the coin must be tossed before the interview, Yuna’s second answer would be correct.) But when she introduces the fact that the coin is still lying on the table, she makes it uncalcuable, because Yuna frankly can’t know how probable it is that Ye Thuza would deliberately let the coin ly there until the next day. Also, Ye Thuza has not even explicitely stated that Yuna should assume everything she told her was true (if not, that would introduce the further factors of how probable it is that Ye Thuza owns such a mind altering substance in the first place, that she would apply it on her child as a punishment and that she would actually let her sleep through a school day.

@ Bartimaeus:Bartimaeuswrote:Given that this follows strips about insane Japanese visual novels involving time travel, I’d say that it’s very possible.

A fun riddle, because it’s trying deliberately to confuse you. That said, once you ignore the confusing factors, the actual answer is rather simple. As it often is. 🙂

Yuna doesn’t know when this coin is flipped and therefore it’s irrelevant to the solution.

All she DOES know is that she wakes up in one of three “game states”, which are impossible to tell apart, but all do have an actual probability.

In 50% of the cases, state Monday + Heads occurs.

In 50% of the cases, states Monday + Tails and Tuesday + Tails occur.

This leads to the conclusion that all three states have an equal 1/3 chance to occur. Therefore there is a 1/3 chance for Yuna that the coin landed on heads and 2/3 chance that it landed on tails.

An easier, less abstract way to picture this is that you perform the experiment 100 times. You have 50x Monday + Heads, 50x Monday + Tails and 50x Tuesday + Tails. Remember thal all Yuna knows is that she cannot tell the three states apart and that each state has a probability. Once you realize that (which can naturally take a while) it’s simple. 🙂

I’ll definitely pose this one to my friends!

The chance of heads given being interviewed is 50%. You can run Bayes Theorem to prove that. Being woken up and interviewed gives you no information about the coin, because you’re always interviewed.

However, if the problem were “guess the result of the flip” with $100 being deposited into her account for being right, then guessing heads would have expected value of 100*0.5 or $50, while guessing tails would be twice that, and she’d be better off guessing tails all the time.

@ Xezlec: Actually we are. There are four iterations here: the first is that the coin landed on heads, and it is Monday, the second is that the coin landed on heads and it is Tuesday, the third is that the coin landed on Tails and it is Monday, and the fourth is that the coin landed on tails and it is Tuesday. Since the Subject does not know what day it is for sure, and the experiment is being carried out, from the description of the experiment, we know that it cannot be Tuesday AND the coin cannot have landed on heads. There fore we have three possible iterations, two of which state that it is Monday and one of which states that the coin came up heads, thus all things being equal there is a 1/3 chance that it is Monday and the coin came up heads.BUT there is a wrench in the works. The experimenter states that THE COIN IS LYING ON THE KITCHEN TABLE, this is more information. We now have an additional possible condition, which is that “the first days experiments were carried out before the coin was flipped” or “the first days experiments were carried out after the coin was flipped” this brings the number of iterations to eight. However the Tuesday results, being independant of the initial state of the coin flip collapse into two equally likely possibilities, resulting in six iterations, one of which is impossible, so the argument could be made that the answer is 2/5ths.

It’s a mind game; Ye Thuza said so. There was no coin flip, no drug, and it’s Monday. The coin on the table is there to keep Yuna off balance, regardless of which way it faces.

Guys, it’s only one coin toss.

And if a coin is still on a table, it’s obviously Monday.

Well thats interesting –

so – lets assume its monday or tuesday… that means according to diagram of the experiment the coind must have been tossed already.

In general there is a miniscule chance that coin will land on the edge, but lets disregard that.

So we have in essence 3 options –

Its monday and the coin landed heads

Its monday and the coin landed tails

Its tuesday and the coin landed tails

where the coin is now is in essence irrelevant.

In general we do know that probability of the coin is 50:50, but I would say we are determining the probability that coin landed heads in this particular experiment is 1/3. At any time Yuna can verifiably call the probabilities the 3 scenarios are equally likely.

@ Paeris Kiran:No it’s not. It’s 1/2. “Monday, landed tails” and “Tuesday, landed tails” are the same scenario, just in different points in time, which is irrelevant for how the coin landed.

@ Invenblocker:Except, how do you come to 75% chance of it being Monday and 25% chance of it being Tuesday? I guess you tried:

P(Monday)

= P(Monday|heads) * P(heads) + P(Monday|tails) * P(tails)

= 1 * 1/2 + 1/2 * 1/2

However, this doesn’t work, it uses the exact P(heads) you’re trying to find, which isn’t 1/2.

However, if you use the correct answer, P(heads) = 1/3, you’ll get:

P(Monday)

= 1 * 1/3 + 1/2 * 2/3 = 1/3 + 1/3 = 2/3

and

P(heads)

= P(heads|Monday) * P(Monday) + P(heads|Tuesday) * P(Tuesday)

= 1/2 * 2/3 + 0 * 1/3 = 1/3

Here’s my take:

Basically, you have four possible configurations which are all equally likely to occur:

P(heads, Monday) = P(heads, Tuesday) = P(tails, Monday) = P(tails, Tuesday) = 1/4

Since Yuna was woken up, you know it’s not heads and Tuesday, but all other configurations are still equally likely. This means the chance the coinflip showed heads is:

P(heads, Monday) / (P(heads, Monday) + P(tails, Monday) + P(tails, Tuesday))

= (1/4) / (3/4) = 1/3

@Cronek :

The coin lands head or tail (same possibility, 50:50)

If the coin shows head (50%), It’s monday when you wake up.

if the coin shows tails (50%), it’s monday or thursday when you wake up (same possibility)

So we have this possibilities:

50% Head/Monday

25% Tail/Monday

25% Tail/Thursday

= 75% it’s monday when you wake up, 25% it’s thursday

On monday it could be head or tail (same possibility), so the 75% split to 37,5% Head/Monday ans 37,5% Tail/Monday. The 25% Thursday all always Tail/Thursday.

= 37,5% Head, 62,5% Tail

I don’t like the question, since it is so convoluted you really don’t know what is being asked.

Here is the question again: “What is your credence now for the proposition that the coin landed heads?”

My credence?! What the hell does that mean?? Credence has nothing whatsoever to do with probability, and yet that was the answer given.

Now, if it had been asked, “What is the probability that the coin landed heads?” The answer is 50%. What is done as a result of the coin flip does not affect the coin flip itself.

But the way it is worded, you have no idea.

@ JBento:I guess we are arguing about the description of “action” to which we assign probability. Beeing in the place of observer of this experiment, you on tuesday can not even determine wheather its tuesday.

so – if you are to base probability on which side of coin flipped based on the only fact that you are awake, wheather its monday and tuesday can not be linked to single parameter. I would not link them as such.

It is not 50%. I think it is 75%-25%, but I’m not sure. What I am sure about, though, is that it’s not 50%-50%

Imagine another scenario, where Thuza re-flips coin prior to waking Yuna up on Tuesday. In this case, the answer is obviously 50%, as every time Yuna wakes up a independent coin was flipped just before it.

Now, imagine that the coin Thuza flips on Tuesday is bias to land on tails. Yuna’s probabilities need to be bias to tails now, as this sceniaro is strictly more “tails” heavy than the previously mentioned one.

At the extreme, imagine the cain that Thuza flips on Tuesday is 100% guaranteed to land on tails. That is the analogue of this scenario.

Now I’m thinking that the Tuesday flip biases the totaly outcome by 25% due to intuition (and imagine you were to change the scenario to be 10000 repeats on a Tails, it “feels” right that the probabilitywould approach 100%)… however, I’m not sure of this.

I ran into this paradox just a few days ago.

Upon some discussion, the issue lies in the vague definition of “creedence.” It is ambiguous whether her level of belief should be the odds that the coin came up heads or the odds that this is an interview after the coin came up heads, given that even though she knows she is being interviewed that wasn’t included in the question posed.

I’m guessing that the coin landed edge.

@ Paeris Kiran:Counterpoint: You have cause and effect turned around. The coin influences the day of the week probability, not the other way around. If the question was referring to the day of the week, then you0d have 75% chance of Monday (because the 50% chance of heads is always Monday, and half of the 50% chance of tails is Tuesday). But the odds of the coin landing on either side (assuming the normal assumptions for this sort of problem, e.g., that the coin is perfectly balanced and that “edge” isn’t a valid result) is 50%.

Neither it is held on its edge In the table.

@ Cervisia:Actually I heard that they recently did an experiment to put that “50/50” odds to the test and discovered that there is actually a slightly greater chance that a coin will come up heads. It lead them to determine that our universe is “slightly positive.”

From Steins gate to Zero escape? Somebody is on a VN kick recently. 😀

@ lucinos:True – see https://en.wikipedia.org/wiki/Sleeping_Beauty_problem#Operationalization; I’m assuming something like that she either remembers all events or forgets all events on Wednesday morning, otherwise you have to treat the days somewhat differently.

@ drs:What if she was only woken up if it were heads? That would give information. The claim you would have to refute would be that waking up different numbers of times on H/T also gives some information.

I mean, granted we agree on how you should behave, so anything else is really a matter of semantics – it doesn’t matter whether you accommodate the consequences via the utility function or via the probabilities. Myself, I take the probabilities to be something like “optimize the number of times I am right,” which would be exactly the sort of situation in which you think I should effectively bet as though the answer was 1/3.

BorgLordwrote:Indeed. Imagine Ye Thuza repeats this experiment 100 times and assume the coin lands on heads half the time.

Yuna would get interviewed 50 times when the coin comes up as heads and 100 times when the coin comes up as tails. So at any awakening, it’s twice as likely that she is being interviewed when the coin landed tails, meaning 1/3 heads and 2/3 tails.

On the other hand, the coin landed heads 50 times and tails 50 times as well. So, the coin landed heads 1/2.

The correct answer depends on which of those two interpretations is the one that you should go for.

@ Shorts:Actually no, as Yuna also gets interviewed on Monday, so it could still be monday. In that case, she will also have amnesia and will have to answer the same question again tomorrow.

So if today is monday then the chance is 50% that it’s either heads or tails. If it is tuesday, then it has to have been tails. But as you don’t know if today is monday or tuesday, you have to take that into account. So there’s basically a 50% chance that it’s monday and 50% chance that it is tuesday. In the second case, you’re actually answering the question again but without remembering your previous answer. So you could argue that there’s a 1-in-3 chance that it’s heads and 2-in-3 chance of tails. (One outcome on tuesday and two on monday makes three possible outcomes. Two of them tails, one head.)

As you don’t know what today is, you should guess tails as that gives you a 66.7% chance of getting the right answer.

@ alanaktion:Actually, it relies on which side is face up when you flip, slightly favoring the upper side.

Magnemawrote:Oh, thank you. That’s very clearly put. I see the controversy now. It’s about what the word “credence” means. I agree that it would not be wise for Yuna to place a bet based on the objective probability, given that the number of bets she places depends on the result. If she wants to minimize the **expected error over all interviews**, then she should use 1/3 rather than 1/2. If, instead, she wants to minimize the **expected error over all coin flips**, then she should use the objective probability, which is still 1/2. Which of those is defined as “credence” is up to you.

@ Xezlec:It’s 100% – or that bitch fed her own kid midazolam, and is about to hear from DFS / CPS / whatever they call “Child Protection Services” where you are!

In order to give an accurate answer, we need a clearer definition of the question. You devious writer you. As the comments have so clearly indicated, interpretations abound.

50%. There’s nothing at all special about whether it’s Monday or Tuesday, since Ye Thuza wakes her up to interview her every night of the experiment.

This is pretty much the same setup, mathematically speaking…Yuna sits on the couch in the living room. Ye Thuza is in the kitchen. On the kitchen table are two bowls, one blue, one red. The blue bowl contains a single tennis ball. The red bowl contains two tennis balls. All three tennis balls are identical. Ye Thuza flips a coin. If it comes up heads, Ye Thuza takes the one tennis ball from the blue ball and brings it to Yuna. If the coin comes up tails, Ye Thuza takes one of the two tennis balls from the red bowl and brings it to Yuna.

I’m pretty impressed. This is actually kind of a brilliant deconstruction of Monty Hall, with like the exact opposite effect on people. My initial reaction was that it would 1/3. But like with Monty Hall, when you take it back to the initial conditions, the answer’s pretty obvious.

Oh that one’s really easy. The answer is… going back to sleep.

The probability can be anything between 0% and 100% inclusive.

The question is what does she think the probability is.

As established by others, her answer is the same no matter when she is woken.

So say she believes the probability of heads is

p.That means that if the coin is heads she believes this happens with probability

p, and the expected value of it being heads isp*p,She believes the probability that the coin is tails is

1-pfor all of the interrogations.The expected value of heads when the coin is tails is then

(1-p)*p.So the final expected value (or the value she believes it is) isp*p+(1-p)*p=p,So if she believes the coin will be heads

p, the expected value works out to bep.So the believed odds are a self consistent value.

Yuna could believe the odds were 10%, and then calculate what the expected odds would be and they would be 10%.

So I believe all answers that have been given are correct.

Of course that also does not really make any sense does it?

There’s a non-magical version of the Sleeping Beauty problem called “The Sailor’s Child discussed at:

http://www.cs.utoronto.ca/~radford/ftp/anth2.pdf

The way I see it is like this:

P(heads)=0.5 because a fair coinflip is 50/50.

I don’t think this is what Yuna needs, she needs P(heads | morning interview), as she knows she is being interviewed. To mathematically find that, we need some other probabilities:

P(morning interview on monday | heads) = 0.5

P(morning interview on tuesday | heads) = 0

P(morning interview on monday| tails) = 0.5

P(morning interview on tuesday | tails) =0.5

So

P(morning interview | heads) = 0.5

P(morning interview | tails) = 1

This gives us

P(morning interview and heads)=0.25

P(morning interview and tails)=0.5

thus

P(morning interview)= 0.75

so

P(heads | morning interview)=0.25/0.75 = 1/3.

For this I assumed the coin was flipped before the monday interview, because of the diagram in panel 2.

To answer the question one must know the probability that Ye Thuza is just yanking Yuna’s chain.

there’s a 50% chance of heads or tails. Half of the time on time on tails it would be Tuesday half Monday. (25% 25%) her telling her its Monday eliminates the chance its Tuesday on the heads leaving a the 50% heads or 25% tails so the odds are 2/3 its heads. The original odds are 50 50 but the information of its Monday eliminates a possibility of tails. 2/3 heads final answer

@ Redenhalter:My claim was that there is not a 75% chance it’s Monday, which you fail to disprove. You could introduce an additional parameter:

M = it’s Monday, -M = it’s not Monday

H = flip showed heads, -H = flip showed tails

W = Yuna was woken up, -W = Yuna wasn’t woken up

You’re really looking for P(H|W) = P(H|M,W) * P(M|W) + P(H|-M,W) * P(-M|W) = 1/2 * P(M|W) + 0 * P(-M|W) where P(H|M,W) = 1/2, because if Yuna was woken up and it’s Monday, there’s an equal chance of heads or tails

Now, P(M|W) = P(M,H|W) + P(M,-H|W) = P(M|H,W) * P(H|W) + P(M|-H,W) * P(-H|W) = 1 * P(H|W) + 1/2 * P(-H|W)

Note that in this last formula, you don’t have P(H) and P(-H) (which would indeed be 1/2 and make P(M|W) = 3/4), but the original P(H|W) and P(-H|W), which are NOT 1/2, you need to find a value so it does work out:

P(H|W) = 1/2 * P(M|W) = 1/2 * (P(H|W) + 1/2 * (1 – P(H|W))) = 1/2 * (1/2 + 1/2 P(H|W)) = 1/4 + 1/4 P(H|W)

multiplying both sides by 4: 4 P(H|W) = 1 + P(H|W) or 3 P(H|W) = 1 or P(H|W) = 1/3

Well there seems to be some serious maths going on, so I hope someone can consider my point and point out the misconceptions simply.

Wouldn’t the probability of day she woke up on change according to the result of the coin toss?

Like, in the first stage, the coin toss result is at a 50% probability for both results.

If its heads, its monday for sure.

If its tails, then the probability that she was woken up on monday or tuesday then becomes 50-50 again.

So… if one were to draw out the probability tree… it would then be [50%-Heads, Monday], [25%-Tails, Monday], [25%-Tails, Tuesday]…wouldn’t it?

@ Pankek:You’re exactly right. To put it another way, two 50% chances happening in a row can be written as (1/2)*(1/2)= 1/4

@ Cronek:You’re conflating the probability of it being a particular day with the probability of the initial coin flip. The coin was flipped, 50% odd each of heads or tails. (An ideal coin, we’re assuming)

Ye Thuza is going to wake Yuna every night that the experiment is active. If the coin landed heads, Ye Thuza would wake Yuna on Monday, like we see above. If the coin landed tails, Ye Thuza would wake Yuna on Monday and Tuesday, like we see above. Each night would be identical, and there’s no way of telling which is which from Yuna’s (and our) perspective.

Look at it from this perspective: If Yuna knew the coin flip was tails, then she’d know there was a 50% chance of it being either Monday or Tuesday, yes? But, she doesn’t know what the coin flip was. So, there’s a 50% chance it was heads, and this is Monday, and a 50% chance that it was tails, and it’s either Monday or Tuesday. Whether or not it’s Monday or Tuesday is an entirely separate question from the initial coin flip, and does not, from her perspective and ours, alter the 50-50 odds of the initial coin flip.

Suppose five hundred thousand worlds where the coin comes up heads, then five hundred thousand interviews where the coin came up heads,

Suppose five hundred thousand worlds where the coin comes up tails, then a million interviews where the coin came up tails.

In a typical interview, probability the coin came up heads is one third, probability it came up tails is two thirds.

@ redneck01:That is a novel way of framing the situation, and it gave me serious pause.

But there is a flaw in that logic. For the odds to be 1/3 (or 2/3, or 3/3 for that matter) there have to be three possibilities. Now, from an outsider’s perspective, it looks like there are three possibilities. As you said, a million worlds, five hundred thousand with one interview (heads), five hundred thousand with two (tails). So if you pick a random interview, there’s a 2/3 chance it’s from a tails world, and a 1/3 chance it’s from a heads world.

But there’s the rub (and I hope I’m explaining this well.) You are standing back and picking from the outcomes of multiple coin flips. In picking a random interview out of three possible interviews, you are assuming two coin flips took place, and one ended up heads and one ended up tails. which gives you the three possible choices. But in each world, there is only one coin flip, so there can never be three interviews to choose from. There can only be one interview, or two interviews.

A man has one ping pong ball in one pocket, and two ping pong balls in the other. (He’s also wearing really baggy pants so you can’t tell which is which.) He flips a coin…if it’s heads, he gives you the ping pong ball from the one ball pocket. If it’s tails, he gives you a ping pong ball from the two ball pocket. The coin flip didn’t determine which of the three balls you got, only which of the two pockets it came from.

50% we must assume it’s a fair coin, and for a fair coin it’s ALWAYS 50%.

Actually, it is just over 50% in favor of the side of the coin that is visible. Ye Thuza never said there would be a second coin toss, only that the coin toss would determine which of the two paths would be taken. She never said there would be another coin toss after the first. This is one of those ploys to make you lose focus.

This reminds me of a similar one I got in university. Imagine you pick one random family with two children. You know that at least one of the children is a boy. What is the probability that the family has two boys?

I won’t post the answer right away, so take your time and think about it.

@ van den Berg:Is it the oldest who is a boy? :p

@ Potatamoto:You’re analogy misses an important point.

To fix that:

On heads, the man first gives you the ping pong ball from his one ball pocket (monday), then he gives you nothing (tuesday).

On tails, the man first gives you one ball (monday) and then the second ball (tuesday) both from the second pocket.

You drop one of the balls (thus making the analogy of being interviewed).

In 1/3 of the cases it’s the heads ball, in 2/3 of the cases it’s one of the tails balls.

Back to the original problem: (also see my previous post)

There are 4 equally likely situations to occur: (with fair coins and equal likelihood of the days)

heads and monday -> 1/4 (Yuna is interviewed)

heads and tuesday -> 1/4 (Yuna is NOT interviewed)

tails and monday -> 1/4 (Yuna is interviewed)

tails and tuesday -> 1/4 (Yuna is interviewed)

BUT Yuna is being interviewed! This means that “heads and tuesday -> 1/4” is not an option in her situation.

This means that heads is (once again) (1/4)/(3/4)=1/3 and tails is 2/3 (for her situation).

The fact that Yuna is awake is information that skews the original 50% probability.

(This of course assumes there are no shenanigans going on that make it impossible to make any sort of prediction (like “it’s all a dream” and stuff))

@ Yannick:Edit: (It’s so easy to get yourself taken away in an argument)

Clearer fix:

On heads, the man first gives you the ping pong ball from his one ball pocket (monday), then he gives you a fake ball (tuesday).

On tails, the man first gives you one ball (monday) and then the second ball (tuesday) both from the second pocket.

You drop something (thus making the analogy of being interviewed).

In 1/4 of the cases it’s the heads ball, in 2/4 of the cases it’s one of the tails balls. (in 1/4 of the cases it’s the fake ball)

So 1/3 of the real drops is heads. 2/3 of the real drops is tails.

@ Potatamoto:With good reason. The chances of the coinflip and the chances of which day of the week it is are intertwined, neither is independent from the other. Consider four possible situations, each are equally likely to occur – supposing a fair coin is used, you have 50% heads and 50% tails, and unless the world suddenly ends, each Monday will be followed by a Tuesday, whether Ye Thuza wakes up Yuna or not)

– It is Monday, heads came up (and Ye Thuza woke up Yuna)

– It is Monday, tails came up (and Ye Thuza woke up Yuna)

– It is Tuesday, heads came up (and Ye Thuza didn’t wake up Yuna)

– It is Tuesday, tails came up (and Ye Thuza woke up Yuna)

Since we know today Ye Thuza woke up Yuna, we can eliminate the third option, but the other three are still equally likely to occur.