The Sleeping Beauty Problem

This famous puzzle illuminates a deep controversy over what we mean by "probability," and how to reason when observer selection effects are in play.

[sullainvictus]

It seems like the thirders seem to think that the question is not "What are the chances the coin came up heads?" but rather "What are the chances that I'm asking you this question under a scenario where the coin came up heads?" Because the question "What is the probability that the coin came up heads?" has nothing to do with you being woken up.

[moosemoss]

I think everyone can agree that they should divide the two camps into the “halves and the halve nots”.

[GarryBurgess]

This seems like a big clue to why the "Many worlds" interpretation of quantum mechanics seems to make the probabilities that we observe more understandable.

[blaavass]

Seems to me that the debate is more about how to understand the question rather than the likelyhood of the cointoss itself:)

[skillzorz]

She's twice as likely to be woken up on the tails flip, but only because she is woken up twice on it. This doesn't change the fact that the coin has a 50/50 chance of landing H/T. The odds of her being awoken twice are equal to the odds of her being awoken once. Both 2/3 and 1/2 are correct, but to two different questions. Just because you record data twice on tails, doesn't mean the odds are higher it will land on tails - just that you're more likely to be recording data during tails flips. Can we all agree on that?

[jimnesstarlyngdohnonglait]

Think more the Hardest way as how Graphic Interchange Format uses 256 colours
Vs
Think more the Hardest way as how it is not correct when an Anchor tag or <A> tag in HTML is used to define tags in a webpage.

[smalin]

The question Sleeping Beauty is asked is "What do you think is the probability the coin came up heads?" Is the question ambiguous? I don't know, but I do know that some of the disagreement arises from people taking the question to be asking different things. Here are some possible rephrasings of the question that have different answers:

Q: What is the probability that a flipped coin will come up heads?

Q: If I flip a coin and ask one person to guess the outcome if it comes up heads and two people to guess the outcome if it comes up tails, which answer will result in more correct guesses?

Q: Is Sleeping Beauty equally likely to be woken up (and asked the question) when the coin flip is heads as when it is tails?

Q: If you run the experiment 100 times, and the coin lands heads 50 times and tails 50 times, how many of the 150 times Sleeping Beauty is asked the question will the coin have landed heads?

Q: If you run the experiment 100 times, and the coin lands heads 50 times and tails 50 times, in how many of 100 trials will heads be the correct answer?

[louiswilbur]

The sleeping beauty problem is ambiguous because it does not say what sample space she is using.  Probabilities are defined on a per sample space basis.  The sample space of the coin toss is {H, T} and the sample space for the questions about the coin state is {MH, MT, UT} where H=heads, T=tails, M=Mondays and U=Tuesdays.  The probability of heads for the first sample space is 1/2 and the probability of heads for the second sample space is 1/3, since they are both equiprobable sample spaces.  To see equiprobability, just notice that out of every 1000 coin tosses about 500 will be heads, 500 will be tails, and about 1500 questions will be asked about 500 which will occur when it is Monday and heads, another 500 which will occur when it is Monday and tails, and the remaining 500 which will occur when it is Tuesday and tails.

She should use the probability for the sample space she assumed and the problem doesn't tell what sample space that is.  The problem is bad because it introduces two different sample spaces without clarifying which one is operative.  For example, if the problem also stated that for betting purposes on repeated trials of the experiment she should bet as much money as possible then it would be clear that she should use the sample space for the questions about the coin state to get the probability.  But if instead of that we added to the original problem that she give the probability for repeated tosses of the coin then money won or lost is irrelevant and she should use the sample space for the coin toss to get the probability.  The sleeping beauty question is ambiguous because it is asking about belief in the frequency of the truth value of occurrences of the PROPOSITOIN "the coin landed heads" not the proposition that the coin's probability of landing heads is 1/2.  That is, the question doesn't make clear if it is asking about the probability of the proposition being true during repeated coin tosses or if it is asking about the probability of the proposition being true during repeated questioning in many repetitions of the experiment.  These are not the same thing because when the coin is tails she is questioned twice but when the coin is heads she is questioned only once.

Now she knows the proposition is true one out of every three times she is asked and she is not going to mistake that for the fact that the coin comes up heads one out of every two times during coin tossing.  So, for the proposition "the coin landed heads" the frequency of this proposition being true during repeated questioning in many repetitions of the experiment is different than the frequency of it being true during repeated coin tosses.  If the coin toss actually came up heads then the proposition "the coin landed heads" is true but if the coin toss actually came up tails then the proposition "the coin landed heads" is false.   How often the proposition is true or not depends on the circumstances.  So adding two different prepositional phrases onto the original question highlights the ambiguity of that question:

(case 1)
What is your belief now for the proposition that "the coin landed heads" in the case of repeated questioning in repetitions of the experiment?
(case 2)
What is your belief now for the proposition that "the coin landed heads" in the case of repeated tosses of the coin?

The conclusion: the sleeping beauty problem is ambiguous because case 1 and case 2 use different sample spaces and if one removes the phrase "questioning in repetitions of the experiment" from case 1 and removes the phrase "tosses of the coin" from case 2 then the ambiguity of the original question is exposed.

[pueraeternus]

The odds of it being Tuesday aren't the same as the odds of it being Monday. When she is asked, she can deduce that the probability of it being Monday with heads is twice as likely as the probability of it being either Monday with tails or Tuesday (with tails). So the correct answer is always 1/2. If you were asking her whether it was heads or tails, the answer would be tails 2/3rds of the time, but that's not the same as the question she is being asked.

[christianblack]

Beaut is not as dumb as mathos and philosophers. She can tell from her leg stubble whether she's been asleep for 1 or 2 days. Drugging women for 'probability research', yeah, right. She's getting a lawyer.

[measureofdoubt]

I can see that a common response is,  "Philosophers don't understand math, it's clearly 1/2."
Sigh. Perhaps I should have clarified that *mathematicians* are divided on this puzzle, too, not just philosophers. For example, here are several mathematicians and physicists arguing the 1/3 position:

1. Nick Wedd is an International Math Olympiad winner, and a thirder:
2. Jeffrey Rosenthal, winner of the "Nobel prize of statistics", is a thirder:

3. Another International Math Olympiad winner, Tanya Khovanova, is a thirder:
4. Physicist Sean Carroll is a thirder (but agrees it's controversial):

[jimh]

What are the "3 possible scenarios?

[Robbyrool]

The confusion can be resolved as follows: It is a question of which timeline she is in, not which day. She was asked essentially what is the probability that you are in a timeline where heads came up. The probability of entering either timeline is 1/2. Once in a given timeline, the probability does not change. What changes is the number of times she will be asked the question. She will be asked the question more times in one timeline. That does not change the correct answer, it only changes how many times she will answer. She will say the correct answer more times if tails came up. But it is the same answer. If she gets a dollar each time she gets it right, she gets more money if tails came up. But the answer is the same. You can safely remove the Monday and Tuesday which changes nothing. If she were asked repeatedly on the same day with a dose of amnesia in between each time, it would boil down to: what are the odds that you will have been asked this question just once vs repeatedly? Answer: 1/2. If the question was: what are the odds that this is the only time you will have been asked this question, the answer would be 1/3. But that would not be analogous to the question in the premise.

[DarthCalculus]

Thinking about this problem, I have a different perspective that clarifies the question for me:
Flip a coin. If heads, take a picture of a dog. If tails, take two pictures of a cat.
Question 1: what is the probability that the coin flipped heads? 1/2
Question 2: choose a picture at random. What is the probability it's a picture of a dog? 1/3

[Eye_Exist]

If the question is Which is the probability the coin landed on heads? the answer is always 50%, no matter if you ask this million times, as increasing the amount of questions on the other side doesn't increase the chance of the coin falling onto that side. The amount of wrong answers if the question is changed _Which side you believe the coin fell on?_ increases by adding more questions on another side if you keep answering Heads, but that is different statistical question entirely. The whole statistical analysis proposed in 1/3 answer is a red herring used to confuse you the half way forgetting what the original question was.

[abojasem]

The answer i believe is 1/2 whether you look at it from an outsider point of view or from beauty's perspective.

Am going to change the experiment a bit just to make easier. Say that if the coin came up head, she would wake up on Sunday. Everything else holds the same. Thus, the question he asks her "what is the probability that the coin came up head? " is the same as "what is the probability that today is Sunday?". This change should not effect the probability at all.

From beauty's perspective, she doesn't know whether she woke on Sunday from the Head scenario, Monday from tails scenario or Tuesday from tails scenario. What she knows is the probabilities of each one. She knows that since the coin is fair, there is a 50% chance that she woke up on Sunday. Also, that there is a 50% chance she woke up on Monday or Tuesday. Therefore, assuming probability of waking up on Monday is the same as waking up on Tuesday, then the probability she woke up on Monday is 25% and probability she woke up on Tuesday is 25٪ too.

Thus, her answer is 50% either way. The mistake i think 1/3 advocaters is that they think the probability of her waking up on Sunday, monday and Tuesday is all the same.

[georgedunn]

I think Beauty's answer would be, in the case of tails, "Coin? What coin? What are you talking about? Who are you?"

[pa-mo]

So it sounds like you would ask her the question every time she is woken up. (If you don't, then she should just answer 1/2.)

If you do ask her the question every time she is woken up, then you have to look at her goal:
- If her goal is to guess right most often on the final answer, then she should still stick with 1/2.
- If her goal is to guess right most often, including when asked on the Tails Monday scenario, then should say 1/3.

[robsmith]

Has anyone noticed the way Julia subtly changes the question when she's explaining the 1/3 idea. She says something like "if the experiment is repeated many times and I find myself answering heads, I'll only be right only one third of the time" but the question wasn't "did it fall heads or tails", it was "what was the proability that it fell heads". If she answers 1/2, she'd be answering right 100% of the time (or 0% of the time if you think the answer's 1/3).

[strideman]

Ha ha, "...Rest assured...". That's my favorite part.