Sleeping Beauty Paradox

Philosophy Forums Science, Technology, and Math
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I recently learned of the Sleeping Beauty Paradox, I found it interesting so I thought I’d share.

Suppose you are part of an experiment: you will be made unconscious, and the experimenter will flip a coin. If it comes up heads, you will be revived at time t_1 and interviewed. If it comes up tails, you will be revived at time t_1, interviewed, then made unconscious again with your memory of being revived erased, and then revived again at t_2 and interviewed again.

Assume that there are no indicia of the passage of time, you only know the experimental setup and that you have just been revived and questioned; that you know the memory erasure to be completely effective and have no side affects; that you know the coin to be fair, etc.

You are revived by the experimenter, who asks you one question: “What is your credence that I have flipped heads?”

What is your answer?

The rest is hidden so you can consider before proceeding.

Discussion

The Wikipedia article gives three possible answers:

1/2 – If it’s a fair coin, there’s a 50-50 chance of flipping heads.

1/3 – there are three possible revivals, and in only one of them would the experimenter have flipped heads.

Mu – the question is ambiguous. Wikipedia offers the following similar problem as a disambiguation:

Imagine tossing a coin, if the coin comes up heads, a green ball is placed into a box; if, instead, the coin comes up tails, two red balls are placed into a box. We repeat this procedure a large number of times until the box is full of balls of both colours. A single ball is then drawn from the box. In this setting, the question from the original problem resolves to one of two different questions: “what is the probability that a green ball was placed in the box” and “what is the probability a green ball was drawn from the box”. These questions ask for the probability of two different events, and thus can have different answers, even though both events are causally dependent on the coin landing heads. (This fact is even more obvious when one considers the complementary questions: “what is the probability that two red balls were placed in the box” and “what is the probability that a red ball was drawn from the box”.)

I lean towards 1/2: There’s a 50% chance that it’s t_1 and the coin landed on heads; a 25% chance that it’s t_1 and it landed on tails, and a 25% chance that it’s t_2 and it landed on tails. I don’t see that being revived gives you additional information sufficient to change your credence that the coin was flipped to heads, your observations are completely consistent with either outcome.

But smarter people than me are convinced otherwise, and the paradox is part of the basis for some versions of the Simulation Hypothesis (because if you take the 1/3 position here, you should similarly think it’s more likely that you’re in one of millions of simulated worlds rather than the one real world).

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Don’t know how to hide this.

record scratch

Hold up.

If there are simulated worlds, they’re in the real world. Unless by real you mean now, and by simulated you mean former/latter to now. Unless of course the sims are mixed/blended.

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It’s really complicated actually. My head reels …

What I can say is:

If Sleeping Beauty finds herself awake, she would realize that she could be in 2 possible timelines: One with the heads and she’s woken up on Monday and the other with the tails and she’s woken up twice, on Monday and Tuesday. She could be in either of these 2 and the probability of heads is 50%.

Also notice that the probability of being woken up is 100% or 1. You can’t do anything with a probability of 1 except multiply or divide with it, which has no effect on the “other guy”. So the probability that the coin landed heads is 50%.

When you count the possibilities like (Monday, Heads), (Monday, Tails), (Tuesday, Tails), take note of the fact that there are 3 possibilities based on DAYS, not the outcome of the coins, of which there are ONLY 2 (heads or tails). You can’t have a probability of 1/3 with only 2 possible outcomes.

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Why not? I can write you a program right now which outputs 1 or 2 randomly, with 1 occurring 1/3 of the time and 2 occurring 2/3 of the time.

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The denominator represents all possible outcomes. If there are only 2 all possible outcomes (with coins, a head or a tail), you can’t have a probability of 1/3. Tertium quid?

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I don’t have any reason to believe this is true, but I have countless reasons to believe it isn’t.

There are certain simple scenarios where this is true, but it’s not some kind of universal truth.

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Apologies, My best hunch is you’re talking about uneven distribution of outcomes. For example if you have a bag with 3 colored balls (1 red, 1 green, 1 blue), you could say a win for you is either a red or green ball and a win for me is a blue ball. Then you have 2/3 chance of winning while I have only a 1/3 chance of winning.

I’m not sure if I’m correct though, but just to complement what I wrote, the tails on Monday carries over to Tuesday (it’s the same tails).

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Yes, probabilities are not always evenly distributed like you’ve been implying.

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a priori versus frequency, right

k byeeeeee

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I have Dm’d you my answer… it may not be right, but I think it very clever and innovative none-the-less. :smile:

[i haven’t read the hidden text and will not do so until I receive your reply]

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.
What has probability got to do with it? How is that supposed to/will that help the subject, in distinguishing between a and b?

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If I know the erasure & have no side-effects, it must not have been completely effective… is that a side-effect? Erasure implies tails was flipped, so.

Zero credence heads was flipped.

Would not submit to such an experiment in the first place, though.

Still. 50-50, a priori. :stuck_out_tongue:

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P.s. Stuff remembered genetically & so instinctual (predispositions, preconditions…)…

…reminds me of an incomplete erasure. Since epigenetics get reset & whatnot.

But not what is transferred like an Olympic torch from adults to children. That is like… epic-memetic (r)evolution, if the cycle-stopper hits reset back to self=other :^P

You can’t pause LIVE, & you can’t erase past or sustainably go further than self=other.

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Apologies to FJ’s thumb’s visceral judgy judgmentalism.

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There’s a kind of poster on forums that likes to drag whatever their particular issue-of-interest is into every thread they can, even when it has nothing to do with what’s going on. They don’t really care what everyone else is talking about, they’re just dying to say their thing, again, the same thing they always say in every thread, regardless of what it’s about.

Your thing is this self=other thing. I’m sure it’s very deep and meaningful and important, but this thread isn’t about that. Nobody’s talking about genetic memory, epigenetics, evolution or self=other in this thread. You don’t need to bring your pet issue with you everywhere you go.

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FJ’s a skimmer!

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This:

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Makes this:

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Click the gear icon at the top of the editing window at the right, there are a few fancier formatting options. Like this → Blur!

You’re equivocating on the word “real”. There’s a sense in which a hallucination is real, but that isn’t the sense I’m using when I make the distinction between a simulation and the real world.

As I said in my DM, it is clever, but answers a different question. This is meant to be an intuition pump of probability/credence, not a puzzle of how to smuggle knowledge. The setup is meant to make knowledge smuggling impossible, and if clever solutions break the setup to allow knowledge smuggling, assume whatever change is necessary to prevent it. Stipulate that Sleeping Beauty doesn’t and can’t know what day it is or what coin was is flipped.

I’ll let you share your response and hold further comment until you do, but for now I’ll share an insight your response suggested.

Consider a slightly different question. Suppose the interview consisted of Sleeping Beauty flipping a coin, and counting whether the flipped coin is the same or different from the coin flipped by the experimenter. How often would the coin be ‘right’ and how often would it be ‘wrong’? Would it be ‘right’ more often when the experimenter flipped heads? I haven’t done the math, but my intuition is that the coin would be ‘right’ 50% of the time, whether the experimenter flipped heads or tails.

Now ask, would a coin with a different likelihood of heads vs. tails be right more often overall? For example, say Sleeping Beauty rolled a fair d6 and picked heads if it came up 5 or 6 and tails if it came up 1, 2, 3, or 4.

If we run the experiment many times, the die roll will be more right than the coin flip, because the die is rolled more often on tails than heads.

I think this is where people who think the answer is 1/3 are coming from. But I still think it’s wrong, basically for the reasons @Hudjefa described:

I think this is right, but it gets at how the question is unexpectedly ambiguous. It’s weird to say that there’s a 1/2 chance you’re on the heads timeline, but only a 1/3 that you’re on the heads timeline and it is Monday. But that seems to be correct.

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So what are you talking about with the simulation? Do you mean like a possible world? Do you mean like a counterfactual universe? You mean like a side virtual thing that’s not part of the main matrix? Because that’s still in the real world. The real world governs anything that is possible. They all (not necessarily all possibilities) happen in the real world. Which is our minds and God’s mind(s) doing stuff together like co-authors except for the fact that we exist in (at the mercy of) the mind(s) of the original author(s)… Who eternally subsume(s) all of time whole and premixed with our contributions. That makes it complicated to call the past hyper & the future hypo, Because not every moment is 100% distinct from every other moment. And I’m not just talking about our remembering a past or future moment in the present moment. Unless I am. In which case it’s just thetical. Like Schrödinger’s cat. So when you say simulation, what do you mean?

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Imagine a highly immersive virtual reality game, where you put on the visor/headphones/suit and all your senses are directly stimulated by electrodes, and the feelings generated along with in-game events are indistinguishable from the sensations you would have from a similar event outside of the game, and the visor/headphone/suit blocks out all other sensations.

Now picture two people, one of them is wearing the visor/headphones/suit, and one is not. The person not wearing the v/h/s is experiencing the “real world”, the person wearing the v/h/s is experiencing a “simulation”.

Note also that while they are both in the “real world”, only one of them is in the “simulation”.

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And how do they know which one they are? Is that what the paradox is about? Is it a brain in the vat thing? An evil demon thing? Like how did the ones who supposedly exited the matrix know they actually exited the matrix?

I have the same answer. It’s all happening in reality. If you live “in your head” the same way you live when you perceive you’re not “in your head” — you are doing real life.

Because even the simulations are in God’s head (reality).

Something I was thinking about earlier that cracked me up (in laughter, not insanity …although admittedly the line is fine …) …

So how long from the “membering” (actual experience of the present/presence) does it become pre-/post-/remembering? What percentage of all versions of that is “membering”— &/or are all versions of that subsumed in “membering”?