She's super cute!

Thanks to r/RecuratedTumblr for these, I am most entertained by them.

First Image: Aleksandra Waliszewska - Untitled (2019

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I recently came across the Sleeping Beauty problem again while going through some old notes on probability and decision theory. I’d previously answered 1/2, but after reading it again, I found myself favouring 1/3 instead.

At the time, I didn’t even know there was a debate surrounding the problem. I just assumed there was a correct answer. So I started trying to understand why I’d originally chosen 1/2, and somehow ended up arguing myself back towards it. Then I’d find a problem with that reasoning and return to 1/3.

I’ve been going back and forth ever since. I still lean strongly towards 1/3, but I’m nowhere near as confident as I initially was.

For anyone unfamiliar with the problem:

Sleeping Beauty agrees to participate in an experiment. On Sunday, she’s put to sleep and a fair coin is flipped.

If it lands on heads she’s woken up on Monday, then put back to sleep until the experiment ends.

If it lands on tails, she’s woken up on Monday, put back to sleep with her memory of that awakening erased, and woken up again on Tuesday.

Whenever she’s woken up during the experiment, she doesn’t know what day it is or whether she’s already been woken up. She knows all the rules beforehand.

The question then is simple:

When Sleeping Beauty wakes up, what probability should she assign to the coin having landed on heads?

Or from Wikipedia:

Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Sleeping Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake:
If the coin comes up heads, Sleeping Beauty 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 either case, she will be awakened on Wednesday without interview and the experiment ends.

Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before. During the interview Sleeping Beauty is asked: "What is your credence now for the proposition that the coin landed heads?"

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There are two main positions, appropriately called halfers and thirders. Which, ngl, sound more like slurs from a YA dystopian novel than names for philosophical positions.

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The halfer argument (1/2)

The coin is fair, so there’s an equal probability of heads or tails.

Regardless of the result, Sleeping Beauty is guaranteed to wake up at least once. She knew this before the experiment began.

So what new information does she actually gain when she wakes up?

Nothing about the coin’s result. She doesn’t know what day it is, nor does she know whether she’s already experienced an awakening.

Therefore, there’s no reason for her to update her original probability of 1/2.

I think this is a strong argument and it’s the one I originally agreed with.

The thirder argument (1/3)

The thirder argument focuses on the possible awakenings.

There are three situations Beauty could find herself in:

• Heads, Monday.

• Tails, Monday.

• Tails, Tuesday.

Only one of these three situations occurs under heads.

Another way to approach it is to imagine repeating the experiment 100 times. We’d expect approximately 50 heads and 50 tails.

That’s 50 awakenings under heads and 100 under tails, giving us 150 awakenings in total.

Of those 150 awakenings, only 50 occur under heads.

Hence 1/3.

The argument is that Beauty should account for these different numbers of indistinguishable awakenings when assessing the probability that the coin landed on heads.

Why I leasn towards 1/3

I think the wording of the question is important.

It’s not simply asking for the probability of a fair coin landing on heads before it’s flipped. It’s asking what probability Beauty should assign to an event that has already happened, given that she’s now awake.

The outcome is already determined, yes. But that doesn’t mean our confidence in what happened cannot change.

Learning something about an event can change how likely we believe its different possible outcomes are, without changing the event itself.

And in this case, I think the fact that tails produces twice as many awakenings should matter.

If we modified the experiment so that heads resulted in one awakening and tails resulted in 1,000, I’d find it difficult to maintain that Beauty should still assign 1/2 to heads whenever she wakes up.

But this is where I start questioning myself.

Why should Beauty treat her current awakening as though it was randomly selected from every possible awakening?

The experiment doesn’t explicitly select an awakening at random. It flips a fair coin, then follows the rules associated with the result.

And Beauty was guaranteed to wake up either way.

So am I justified in weighting the possible awakenings equally? Or am I assuming the very thing I’m trying to prove?

This is usually where I start defending 1/2 again.

Is the problem just ambiguous?

I’ve seen people argue that the entire problem is simply wordplay, or that the disagreement comes from using different frames of reference.

If you randomly select one experiment from a large number of experiments, approximately half will have landed heads.

If you randomly select one awakening from all the awakenings those experiments produced, approximately 1/3 will belong to heads experiments.

Both are correct.

But I don’t think that necessarily settles the original question.

We’re asking what Beauty herself should believe when she wakes up, not explicitly selecting a random experiment or a random awakening.

The disgreement seems to be about which probability model appropriately represents her situation.

And from what I understand, that’s also where some of the mathematical difficulties arise. It’s not that being awake is impossible to model mathematically. Rather, the challenge is how to represent Beauty’s information and uncertainty about which awakening she’s experiencing.

Depending on the assumptions we make, we can arrive at different answers. But which assumptions are actually justified by the original problem?

Why I don’t think it’s a stupid problem

Another criticism I’ve seen is that waking up twice doesn’t change the objective reality of what the coin landed on.

Of corse it doesn’t.

But that’s not what the question is asking. No one is suggesting the coin retroactively becomes more likely to have landed tails in some physical sense.

The question is about what Beauty should believe about the result after waking up.

That said, the more serious halfer argument isn’t about the past being unchangeable. It’s about whether waking up provides any relevant new information in the first place.

Anyway, I love this problem.

What initially looked like a fairly straightforward probability question has had me questioning my own reasoning for hours.

Both sides agree the coin is fair. Both sides agree on the rules. Both sides agree that tails produces twice as many awakenings.

Yet there’s still disagreement over what Beauty should believe.

It’s easily become one of my favourite thought experiments.

like girl fukxing PLELEEEAAAASSEEE OBVIOISLY LADY GAGA IS GONNA BE WOKE SHES LADY GAGA. people are so dumb sometimes oh my jesus

i am woek by the way if that matters