Sleeping Beauty Problem Followup

[hypersonicpiano]

Good explanation. Now we can simplify between the coin and ball scenario to see which interpretation is correct to the original question.

In my opinion the ball analogy does not match the problem because you select only one ball and treat each ball as an independent event. If the coin lands tails Sleeping Beauty wakes up twice. In other words she selects a red ball from a bag with 100 balls (50 blue and 50 red), puts it back and if it's red selects another red ball from a sack exclusively of 50 red balls.

The problem here is due to the memory wipe, which ball selection should we count? The first time she drew a red ball or the second time? I guess this is why the sack has been combined, but in a world where the coin was flipped 100 times I canmot wrap my head around how we get 150 total independent results from a binary choice. There are only 100 "worlds" where the coin gets flipped, wiping Sleeping Beauty's memory does not magically turn 100 worlds into 150 worlds even if it may seem that way from her perspective.

[brianmacker]

Exactly my solution immediately upon hearing the question. The question is ambiguous between two different questions.

[eternalcoldweather]

If you flip a coin, you are assuming at the fall that the answer is ½. But if you flip a coin, and things happen in the real world over 2 days or more, it's not the same question, because there might be a lot of events such that from your point of view, your experience now drastically changes the question, in this case, how. many times you were awoken, instead of instantly knowing the probability.

[mattias]

Maybe I lack understanding, but I can´t see how this resolves the problem. The problem is very clear: The sleeping beauty is asked what she thinks about the coin being tossed, whether it came up one way or another. There is no way that she can give two answers. This intent to find a solution just tries to find away around the question to my understanding. The challange remains: How should she answer the question.

[LoreFriendlyMusic]

The probability that the coin landed tails given that she is awake from her perspective = 50/50.

You can try this your self. Have a friend flip a coin and not tell you what the result is. Then have him write three notes. One that says "given that you see this note, what do you think is the probability that the coin landed on: insert heads if coin landed heads, tails if coin landed tails?" And two notes that say: "given that you see this note, what do you think is the probability that the coin landed on insert heads if coin DIDN'T land heads, tails if coin DIDN'T land tails?" Then have him burn one of the duplicate notes and show them to you. Now pick one.

50/50.

[deghgggd]

I commented on your older video defending this and I just now found this one, I am glad we are in agreeance as you seem very intelligent and respectable.

[udhessi]

Wrong. The question to be answered by SB is the following : "knowing the experimental protocol and knowing that you are awake, but not knowing what day it is, what would you say is the likelihood that the result of the coin toss was tails ?"
The answer to this question is rather straightforward : it's neither 1/2 nor 2/3, but 5/8 (see the discussion of the wikipedia page on this "paradox" for the demonstration.)
Asking a different question, which might have a different answer, is not relevant at all.

[dan]

I watched the original video again, apparently he qualifies the situation as 3 events (Mon head, Mon tail and Tuesday tail), at 2:20. Derek was saying if the coin was flipped before she wakes up scenario. So to me, Derek was asking the question about fair coin, which is 1/2. Just my opinion, it should be 1/2.

[danorris]

You are not allowed to change the question to provide an answer. Period.

What are the odds a coin came up heads? It doesn't matter where you are, what day it is, what color my dog is, what my buddy's ex wife said after she made breakfast, etc.

It's 50/50. Period. If you flip a coin right in front of me and regardless of results you ask, "What is the probability the coin would have come up heads, " it'd STILL be 50/50. It doesn't matter if it's on tails of not. The odds were, are, and will be the same.

If you think the coin having landed on heads or tails matters to the probability of the event you still don't understand the question.

[YannisMar]

In the proposed balls selection alternative there is no justification for mixing all the balls together and then randomly selecting one. Since balls correspond to awakenings, in order to treat all of them as independent events and put them in the same bag for randomly selecting one of them implies that in each trial resulting to a Tails toss only one of the awakenings is selected. Thus, the equivelant ball selection setup given the SB problem setup is to flip a fair coin and in case of heads put a blue ball in the bag ( Monday Heads), whereas in case of tails you should flip another fair coin to put in the bag either a red ball (Monday Tails) or a purple ball (Tuesday Tails). Thus, in the end you will have in the bag about 50 blue, about 25 red, and about 25 purple. Then when you randomly select a ball you have about 50 % chance to get a blue ball (Heads toss) and about 50% to get a red or a purple ball (Tails toss). Notice that this selection is what SB should use to model her uncertainty upon awakening on the day of the week, i.e. if it is Monday and Heads (50%) or Monday and Tails (25%) or Tuesday and Tails (25%). Therefore, it is not a matter of interpretation of what it is asked, SB should always infer that p(Heads) = 1/2.

[kyjo]

The Sleeping Beauty problem is very _clearly and purposefully_ asking the latter question though, not the former. Clearly she is NOT being asked about fairness of the coin flip. She's being asked about probability that is condintional on her being awake.

The very fact that she is now being asked the question gives her additional information compared to before the experiment started, specifically that she is not on Tuesday after the coin was flipped heads - because then she would be asleep and couldn't be asked the question. This update changes the conditional probability of heads flip from 2 out of 4 to 1 out of 3.

[NR.nrnr]

This exact solution was made by Groisman in 2008 in the article 'The End of Sleeping Beauty's Nightmare'.
Arguing that there is no paradox still has it's consequences though, especially in terms of how we should consider scientifical questions usually thought to have objective answers.

[Strtsphr]

*Consider another experiment:*

A coin is flipped and if it is Heads you are directed draw a marble from bag A that contains 5 White marbles and 5 Black marbles. If it is Tails you are to draw a marble from bag B that contains 9 Black marbles and 1 White marble.

The experiment is run: The coin is flipped and the result of the coin flip is concealed from you. You are presented with a bag and directed to draw a marble from it. You know that the bag presented to you is either Bag A or Bag B but since the result of the coin flip was concealed from you, you are unsure which bag you are drawing from.

After running the experiment you ended up with a Black marble. What is the probability that Heads was the result of the initial coin flip?

*Discuss.*

[Marco_Viva]

No! Just no! You do the experiment one time. 50% of the time, you have a blue ball in the bag. 50% of the time, you have 2 red balls in the bag.
If you send sleeping beauty to sleep 100 times of course, it's different. But that is not the experiment. The experiment is to do this whole ordeal one time and only once.

So after the experiment is over and you reach into the bag, get the ball, it's 50% of the time ble and you have no other ball and it's 50% of the time red and you have another red ball in the bag.

Why is this philosophical experiment even a thing. it's 50% and that's it....

[sjoerd]

It is a literacy problem, not a statistics problem, and not a paradox. Yes, the question is being interpreted in two different ways and the answer to the different questions are different. In the context Sleeping Beauty is being woken up, the answer is 1/3. If that is not liked, then ask the question clearly and unambiguously.

Sleeping Beauty knows neither the day nor the how the coin came up. She knows that she will be woken up 2 times for a tail and 1 time for a head.

[kaizokujimbei]

1. Assign an outcome on the result of a flipped coin.
(Heads will have SB wake up once on Monday. Tails will have SB wake up twice on both Monday AND Tuesday.)

2. Define the coin as being fair, meaning it has a 50% chance to land heads and 50% chance to land tails.
(This is the answer to the question, "What is the probability SB woke up because of a heads coin flip?")

3. Execute the pre-decided outcome for the coin flip.
(This is Derek's video. The coin lands on tails, therefore, SB will be awoken twice on both Monday AND Tuesday, and each time she is awoken she will be asked the question "What is the probability that you woke up because of a heads coin flip?" to which her answer will be, assuming that she is intelligent enough to understand how probability works, "It is 50%." Considering that the coin landed tails she will be asked two times the same question. And both of those times, assuming she is intelligent enough to understand how probability works, she will answer the same way. Because why would she decide to answer any different way when she knows the rules of the game before she starts playing it? Why would she answer any other way when she knows the coin is fair? And why would it matter if she gives the correct answer or the wrong answer? Her answer, correct or wrong, will NOT affect the probability that she awoke as a result of either heads or tails anyway. Because it is the the coin that which determines the probability, NOT her opinion of it.)

[RobRoss]

Great explanation! Makes sense to me. You’d think philosophers would be more careful in asking very specific questions. 🤔

[kaizokujimbei]

The initial question Derek asked BEFORE he change it is what is the probability that she woke up because of a heads coin toss? And the coin is flipped ONLY ONCE before we wake her.

Derek is a MANIPULATOR. He took advantage of certain people's inability to pay attention to subtle changes of the question in order to deliberately confuse them. DON'T FALL FOR IT!

The true question is and always will be the first one. Everything that comes after is obfuscation of the facts. This is reflected in Derek's title of the video which automatically updates with the like/dislike ratio. The guy is making fun of us IN REAL TIME!

[selsickr]

Yes it all depends on how you define the probability that we want. Both are as you said just 2 different problems. I think that we must be more practical in the answer and consider more what kind of a case each way would be useful for.


They are describing 2 different things and are useful in 2 different ways. The paradox comes from the fact that we are confusing 2 different cases and saying that they are the same. It is like 1+1=2 and 1+1=10. Both are true. The first is true for arithmetic in base 10 and the second in base 2. In to context of buying apples I will have to pay for 2 of them, in the context of computer calculations it is 10.

If we want to know the chance that we will get heads if we flip the coin it is obvious that the rest of the story about sleeping beauty is not relevevant. She was told that it was a faire coin and therefore it is obvious that the chance that one gets a head is a 1/2. The rest of the story is just a decoy and diversion. It is like the children’s riddle where they tell someone that they are the driver of a bus that has 20 people in. After telling them that 5 people get of and 6 get on etc. they are then asked what the name of the driver is. This way would be useful if sleeping beauty was told that she has to pay 1$ for a lottery ticket and would get 2$ back if she could correctly tells us what the probability is of the coin falling on H is. By saying 1/2 she would win 1$. ( if the coin was not fair and we were told the probability of heads we could also work out a lottery like in the second case)


The second way, which is a different problem is that when she wakes up she has to pay 1$ for a lottery ticket and will get 2$ back if she can tell what was the result of the coin flip before she woke up. Here, if she says tails she will win on average (if this is done many times )
2/3*2-1=1/3$
If she would have said H
1/3*2-1=-1/3$
ie she would lose 1/3$ every week.

The 3 possibility is that sleeping beauty is writing a college exam after her long sleep. Here she would be careful to give the answer that her professor described in class with all the confusions included. If she was too clever her professor might not like her answer and she would not get her grades.

[antog]

At least someone who reconsider his position exist, maybe it's possible to have a sane conversation. So, i would ask you: do you think "What IS the probability CAME up Heads?" is the same of the "what WAS the probability TO COME up Heads?" question?
And then i would ask "if yes, in what form the question has to be asked, in the shortest form possibly, to have a conditional probability problem (where the answer is 1/3...)?".
Thank you.