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Comment by Bartweiss | Hacker News Reader

I'm having trouble finding the paradox here. As with many probability puzzles, the problem seems to be a hidden conditional in the probabilities.

There's no change in belief happening, only two different beliefs. First is "1/2 of the times it is flipped, the coin will be heads". Seconds is "1/3 of the times I am awakened, the coin will be heads". The coin flip is fair, but the decision to ask the question is biased.

This is extremely simple to demonstrate by resorting to the absurd case. If we change the awakening ratio away from 1-2, the absurdity of saying "1/2" becomes increasingly clear. At 1-9, tails will be the correct guess 90% of the time the question is asked. At 0-1, heads isn't even a possible outcome on awakening.

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Both answers are valid because the question is ambiguous. Asking "what is your belief" doesn't work because there are multiple ways to interpret that. You can clarify such situations by putting them in a game. Probabilities are about how much you would bet on it.

Game 1: You get $10 every time you guess correctly. Best strategy: Guess tails because if it's correct you get $20. If you guess heads correct you get only $10. If you got $3 for guessing tails correctly and $6 for guessing heads correctly then it would not matter what strategy you used.

Game 2: If you guess correctly then you get $10 by the end of the experiment. Best strategy: it doesn't matter what you guess, if you guess tails correctly you get the $10, and if you guess heads correctly you also get $10.

Thirders think the question is like game 1: every time they guess correctly they win correctness points. Halfers think the question is like game 2: they either guess correctly or they don't; you don't get additional correctness points by guessing right twice.

Your Game 2 is poorly defined. What does it mean to "guess correctly" when you're asked to guess twice? If you systematically take the first (or the last) answer, then it's equivalent to being awaken only once on Tails. If you take the logical OR of the correctness of two guesses, then the sleeping beauty can win 5/8 of the time by answering randomly, which is better than 1/2 by guessing a fixed answer (the only strategy you analyzed).

Right, but I didn't want to complicate the explanation. Only deterministic strategies are allowed, or require both answers to be correct in order to get a reward.

> Game 2: If you guess correctly then you get $10 by the end of the experiment.

The question is clear you get a reward each time you awaken, not at the end of the game.

The reward is correctness (Humans like being correct = reward, else things like 'should believe' becomes meaningless hence the question becomes meaningless Q.E.D. it's not meaningless and correctness is a reward)

"When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads?"

This fits with my understanding of what's happening here. The SO discussion is a bit unclear as you go down the page, but the initial statement of problem up top clearly claims that you're questioned each time you awaken.

I think you're quite right to point out that being correct is the only fair heuristic for this question. Several of the defenses of a 'halfer' belief amount to "you ought to believe this so that your beliefs obey this type of decision theory". Actively believing something with a (reliably) worse payout to conform to a rule about belief seems like a terrible form of epistemology.

On a related note, are you familiar with Newcomb's Paradox and the LessWrong debates over it? They get to a similar idea of "correctness above all".

For game 2, flip a coin and guess what is shows.

If the original coin was heads, your chance to win is still 50%.

But if the original was tails, and you only have to guess once correct, your chance to guess wrong twice is only 25%.

So a 50% chance of winning half the time and a 75% chance half the time gives you an overall better chance that the 50% of sticking to one answer because it doesn't matter.

It seems a random guess is the best solution for game 2.

For 1-9, tails being the correct guess 90% of the time still makes sense to me.

If you run the experiment twice, typically the first time you get heads, you are woken once, and so heads is correct answer once, the second time you get tails, but you are woken 9 times, so tails is the correct answer 9 times.

Even from the frame of reference of the experimenter, heads and tails are equally likely, but it if you guess heads incorrectly, you will be wrong 9 times, or 9 times as wrong ;).

Basically it depends on what you consider to be a "trial" of the experiment, if a coin toss is one trial, the wakings are a red herring, the probability is the same as the coin toss. If one trial is one waking, 90% of wakings are going to be tails.

The phrasing of the question is When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads?

It's not actually asking for a strategy about guessing the correct answer, or how many times you'll be right or wrong.

It think whether you are a halfer or a thirder is going to depend on how you interpret the question.

If you see it as asking about the frequency at which "Heads" will the right answer, then it's clearly 1/3. But that's not what the question actually asks (probably - all language is interpreted).

"1/3 of the times I am awakened" never happens though. You're either awakened once or twice, not 3 or 6 or infinite times. I think the thirders get stuck on this "if we do it infinite times" or "in the limit case" thing, which is different from doing a single coin flip. http://stats.stackexchange.com/a/169582/87304

I think the problem is in the phrasing of the question: When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads?

That's not a precise probability question.

More significantly the question (whatever it might be) is potentially asked more than once.

So, if the question is "what is the probability that the coin-flip was Heads?" (which I think it the most likely interpretation) then it's 1/2

But if the question is "what is the probability that you are being woken up because the coin was Heads" then it's 1/3

Asking me what I "should believe about the outcome" is (intentionally, I suppose) blurring the question so that it disguises the fact that there are actually 2 different questions in play.

No, those questions are equivalent. In the single experiment case, the answer to both is 1/2, and in the infinite case, it's 1/3. There's no "3" to even enter into the equation in the single experiment case.

Uh? The outcome probability is the same if you do it once or if you do it infinitely, it's an independent event.

The event isn't just "there was a coin flip". The event was "there is a coin flip and I have been woken up". For this event, two thirds of the time the coin flip will have been tails.

Halfers forget that there is an extra piece of information available "I have been woken up". SB knows that one third of the time she has been woken up it is because the coin flip was heads, and two thirds of the time it was tails.

Anyway, this is a stupid "paradox". It is extremely simple to build a simulation of SB being woken up and coin tosses, and if you do so, you get the clear answer - 1/3. There is no sampling issue or other trick that makes this simulation hard to write correctly...

Which answer is "correct" depends on your interpretation of the question, as you say, and that's exactly the point. Like many paradoxes of the kind, it illustrates the ambiguity of the question and, by extension, the lack of nuance in the way we describe and reason about probabilities (in general, that is; of course one who has learned some probability theory is likely to be able to understand and communicate the difference).

Implicit in any answer is a way of valuing the choices made. In the 1/3 case, it is assumed that being right twice is better than being right once - for example, you get $100 each time you answer correctly. In the 1/2 case, it's assumed that answering the same question correctly twice isn't inherently better than answering it correctly once - for example, you have to debate an opponent at each awakening. You're either right or wrong and you don't get extra credit for winning the same debate with the same opponent twice.

Exactly this. I came to make the same comment, but unsurprisingly someone has beat me to it. In particular, the answer to the "paradox" in a nutshell is this:

> There's no change in belief happening, only two different beliefs. First is "1/2 of the times it is flipped, the coin will be heads". Seconds is "1/3 of the times I am awakened, the coin will be heads".

Yep it's not a paradox, it's actually just two separate questions.

1. What is the probability that the coin toss was heads (as posted in question)

    set of possibilities: { H, T }

2. What is the probability that you have awoken due to the coin coming up heads? (implied by question, but not explicitly asked)

    set of possibilities: { H, T, T }

The question is actually: "given that I have woken up, what is the probability of heads?"

This is the question that is disputed by sources in OP. Both sides are addressing the same question.

I agree on the 0:1 case, but say it's even 1:100000000. Before you go to sleep, you're asked "what do you think the result of the coin flip will be?", and then you wake up and are asked "what do you think the coin flip was?" What new information have you received that changed your mind?

The new 'information' is the number of times you're being woken up. Even though this information is not being told to you, you can implicitly use it by using the same strategy every time.

You should see it from the perspective of every awakening.

Imagine this. The experiment is repeated 10 times. On average the coin will have landed 5 times on heads, and 5 times on tails. This means that you'll be woken up 5 times while the coin landed on heads. However, you'll be woken up 500000005 (!) times while the coin landed on tails.

Knowing this, when you're woken up only in 5/500000010 of the scenarios the coin landed on heads.

The fact that no new information was gained does not mean that your answer to "what do you think the coin flip was?" will neccessarily be equal to "what do you think the result of the coin flip will be?"

All it means is that you can already, before going to sleep, make an unconditonal strategy "when woken up, I'll say this". And your knowledge can easily lead you to a statement "right now, I believe the coin flip will be 50/50, but when I wake up I'll bet on tails".

The coin flip does change the probabilities significantly after you get awakened - e.g. in the 0:1 chance you know that being awakened implies that it must be that particular result, and the 1:100000000 scenario implies that if you're awakened then it either was tails or something extremely unlikely happened. The "no new information" clause simply means that you get already informed about this probability before going to sleep, not after the coin flip.

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