What is the Sleeping Beauty Paradox?

An explanation of the Sleeping Beauty paradox for probability. Specifically it is a paradox for Bayesian Epistemology. The video looks a the views of David Lewis and Adam Elga, as well as the Rip Van Winkle paradox and the snake eyes sleeping beauty paradox.

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Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, The Oxford Companion to Philosophy, The Routledge Encyclopedia of Philosophy, The Collier-MacMillan Encyclopedia of Philosophy, the Dictionary of Continental Philosophy, and more!

[CapnSnackbeard]

You should assign a 100 percent probability that the "prince" is a creep.

[Pfhorrest]

Something seems very confused about the notion of this being paradoxical. When the scenario was first described, my initial thought was that you should of course guess that the coin landed tails, because 2/3 of the times you are awoken will be times when the coin landed tails. That is not a guess about *the fairness of the coin* though. It's a guess about the product of the probability of the coin landing one way and the probability of being awoken given the coin landed some way.

Consider for comparison another scenario. Your friend is going to go into another room and flip a coin. If it lands on heads, he will come out and ask you to guess how the coin landed. If it lands on tails, he will flip it again, and repeat that procedure until the coin lands on heads. You know this ahead of time. Now your friend goes into the other room, and comes back out and asks you to guess how the coin landed (on its last flip, natch). You're obviously going to guess "heads", because you know that you will only ever be asked the question if the coin lands on heads. But that doesn't mean you think it's a coin that always lands on heads. "What are the odds of this coin landing on heads per each flip?" and "What are the odds that the last flip before I asked you this question landed on heads?" are different questions, with different correct answers.

Assuming a fair coin, it's a 50/50 chance of it landing on either heads or tails. But that's not what you're being asked about; you're being asked how the last flip of the coin before you were awakened landed. Since you will be awakened twice as often for tails as for heads, the answer to that is 2/3 tails. But the coin still has 1/2 odds of landing tails per flip.

[chrisstott]

Any claim that knowledge of the consequences of a coin flip ought not adjust my assessment of the odds of the coin position, is incorrect on its face. For instance, if I know "The Prince never wakes a sleeper on Tails", I ought name Heads.

[jackychen]

The way I see it:
- The coin landing on a heads or tails is still 1/2 each, there's no doubt about that. The probability that I wake up on a Monday is 2/3 and on a Tuesday is 1/3. The probability I wake up on a day where the coin landed on heads is 1/3 (and 2/3 for tails).
- The big issue for me is what is this paradox asking for? The coin's probability for either outcome is clearly 1/2. The probability of me waking up on a day where the coin landed heads/tails is also clear (1/3 and 2/3 respectively). These aren't mutually exclusive, I just don't get what the question is asking for. If it's just asking for the probability of the coin landing on tails, then 1/2, but if it's asking for the probability that I wake up on a day where it landed on tails, then 2/3.
- Lets view this as a game of sorts where I get 1 point for guessing correctly. The question then becomes "if it lands on tails and I wake up twice, do I get 2 points for guessing tails each time I woke up?" If so, then the whole "what probability should you assign it" makes more sense and has a clear answer. If tails lands me 1 point if and only if I guess tails on both days, then the answer is also clear (as in it's basically the same 50/50 if I always choose one answer). The issue is that the value of an answer in the events hasn't been clarified yet.

[Thendquincy]

This has always been my favourite thought experiment. I learned about it from a video game called Zero Time Dilemma, and it's fascinated me since then

[konnorporter]

The problem with this set up is that of the three events, one event produces another. Let me explain:
P(Awoken on Monday after heads):50%
P(Awoken on Monday after tails):50%
P(Awoken on Tuesday after tails):50%

This is fine because being awoken on Monday and Tuesday for tails are 100% dependent on each other. P(A or B) = P(A)+P(B)-P(A and B)

So, in any run, you will be awakened on a monday for 100% of the runs and tuesday for 50% of runs.
Now these are the probabilities of the general outcomes, not the specific instances, for which being woken up on a monday and tuesday on tails become mutually exclusive. So the probabilities become:
P(Heads and it is Monday)=50%*100%=50%
P(Tails and it is Monday)=50%*50%=25%
P(Tails and it is Tuesday)=50%*50%=25%

Adding up the Probabilities still show that there is a 50/50 probability unchanged!

However, going by pure probability may show the chance of being right, it does not show the full outcome of being right. Remember that if it is tails, you are awoken twice! This means that you can be right twice in a single go if you chose tails!

What this boils down to is that you have a greater expected value from tails as:
E(H) = 1 time correct * 50%= 0.5
E(T) = 2 times correct * 50% = 1

TLDR: The probability for heads or tails is still 50/50, but guessing tails gives a larger payoff (2x)

[jeremyhansen]

Doesn't the 2/3rds response assume that the three options are independent? Given that waking up on Tuesday is very much a function of whether or not you flipped heads or tails, this doesn't seem to be a good assumption to make.

[RogerWKnight]

Prince wakes up Sleeping Beauty. He asks her "How did the coin land when I flipped it?"
She responds: "Who cares. I'm awake now, and well rested. If it was true love's kiss, you wouldn't be asking me stupid questions."

[Aweman]

It seems that the odds of coin aren't changing but rather the value of the wager. The problem could be rewritten to say that two people are betting on a coin flip if both people agree that if the coin lands on tails the value of the wager is doubled, would it be smarter to bet heads or tails. It would obviously be smarter to bet on tails but not because tails is more likely but rather because the consequences are doubled if the coin lands on tails.

[weareallbornmad]

Yeeaah, I still don't get the probability theorem... but got a very striking picture of just how REALLY, REALLY, *REALLY* CREEPY are the fairytales we tell our daughters since before they can walk. Good education, thanks.

[justas]

This kinda reminds me of the "Judge says the prisoner will be executed and will be surprised about what day he dies" scenario where the premise boggles the actual point it's supposed to illustrate.

[cliffordhodge]

I have always been skeptical of the use of statistical analysis applied to a single event (There are lies and damn lies, and then there are statistics). It often depends on some sort of conflation of objective facts and psychological tendencies, but more important, it proceeds from a casual mapping of a manifold, a multiplicity of events (the experimental sample, so to speak) onto a unit - the one event in question. It assumes that the state of affairs regarding the previous events - which state of affairs is a static entity - is, as you might say, convertible into a dynamic entity, namely the one actual event about to unfold. Further, it ignores that inconvenient fact that a single event has a probability of either one or zero - it either occurs or not, but never (for macroscopic events, at least) lands in that imaginary area where the probability p is such that 0<p<1.

[EterPuralis]

Funny how they're always telling the truth in these scenarios. Surely one of the variables should be whether or not the creepy prince will lie about the coin toss so he can keep you in bed indefinitely.

[davidplowman]

What happens if you refuse to answer? Will he only put you to sleep upon a wrong answer? Also, what about cop out answers like the side of the coin that was showing was the one your eyes noted it to be?

I’ll admit I don’t really understand the idea this example is trying to explain, but it seems to me that if the coin was flipped only once you could decide to answer tails on Monday and heads on Tuesday. I also agree with a comment below: philosophy and statistics can be strange bedfellows.

[dojelnotmyrealname]

I mean, real talk, just say Tails. Under those circumstances, just bite the L of being right exactly 50% probable of being right. There is no consequence for failure for saying Tails and being wrong.

[weareallbornmad]

...well this was more mathematics than I was prepared to handle today.

[ajhieb]

Something, something, Monty Hall Problem.

[kyjo]

The fact that you are awake and being asked the question means that it cannot be Tuesday after heads - which is new information compared to before the experiment started.

[neco]

Hold up its a probability of 3/4 heads. On Monday you have a probability of 1/2 (as normal to awake) otherwise another 1/2 on Tuesday times the already existing 1/2 for having tails the day before this is (1/2 + 1/4 ) = 3/4
Furthermore then the Rip Van should be 1/2^(1000) tails witch is a 1, 302 zeros behind the coma. A little more than 1/1000

So either
1. Philosophers are bad at math
2. I am bad at maths
3. Or I just didn't understand the question/scenario correctly

If 2 or 3 please explain

[rosiefay]

0:32 The paradox relies on Sleeping Beauty forgetting about Monday's interaction if the coin came down tails. I do not accept that this is necessarily true.