The Most Controversial Problem in Philosophy

Do not hit the like button! Or the dislike button, at least not yet. I want you to consider a problem that's been one of the most controversial in math and philosophy over the past 20 years. There is no consensus answer. So I want you to listen to the problem and then vote for the answer you prefer using the like and dislike buttons.

Okay, here is the setup: Sleeping Beauty volunteers to be the subject of an experiment. And before it starts, she's informed of the procedure. On Sunday night she will be put to sleep. And then a fair coin will be flipped. If that coin comes up heads, she'll be awakened on Monday and then put back to sleep. If the coin comes up tails she will also be awakened on Monday and put back to sleep. But then she will be awakened on Tuesday as well and then put back to sleep. Now each time she gets put back to sleep she will forget that she was ever awakened.

In the brief period anytime she's awake she will be told no information. But she'll be asked one question: What do you believe is the probability that the coin came up heads? So how should she answer? Feel free to pause the video and answer the question for yourself right now. This was my reaction after hearing the problem for the first time. I mean the intuitive answer that pops into my head is clearly one in three. It could be the Monday when it came up heads. Or it could be the Monday when it came up tails. Or it could be the Tuesday when it came up tails.

But you know what's really interesting is you just answered that the probability of a coin coming up heads is one-third. I think a lot of this comes down to what specific question is asked of her. What is the probability that a fair coin flipped gives heads? That's 50 percent. What is the probability that the coin came up heads? I would say the answer is a third from her perspective. Yeah, it's remarkably the same question.

The simple reason why Sleeping Beauty should say the probability of heads is one half is because she knows the coin is fair. Nothing changes between when the coin is flipped and when she wakes up, and she knew for a fact that she would be woken up, and she receives no new information when that happens. Imagine that instead of flipping the coin after she's asleep, the experimenters flip the coin first and ask her immediately: "What's the probability that the coin came up heads?" Well, she would certainly say one half, so why should anything change after she goes to sleep and wakes up?

This is known as the Halfer position. But there is another way to look at it. Others would argue that something does change when she's awakened. I mean, it seems like she gets no new information. There are no calendars, no one tells her anything, and she knew that she would be woken up. But she actually learns something important.

She learns that she's gone from existing in a reality where there are two possible states: the coin came up either heads or tails to existing in a reality where there are three possible states: Monday heads, Monday tails, or Tuesday tails. And therefore she should assign equal probability to each of these three outcomes where heads only occurred in one. So the probability that the coin came up heads is one-third. This is known as the thirder position.

Now, I know it seems wrong to suggest that a fair coin should have a one-third probability of coming up heads, but that's because the question she's asked is subtly different. The implied question is: "Given you're awake, what's the probability that the coin came up heads?" And that is one-third.

Now halfers would counter that just because there are three possible outcomes doesn't mean they are each equally likely. In the Monty Hall problem, for example, the contestant ultimately has to choose between two doors. But it'd be wrong to assign them 50/50 odds. The prize is actually twice as likely to be behind one door than the other.

In the Sleeping Beauty problem, we know a heads outcome and a tails outcome are equally likely. So the chance of waking up on Monday with heads is 50 percent. And the chance of waking up on Monday or Tuesday with tails should be 50 percent. Therefore the tails probability gets split across two days, 25 percent each. But if you repeat the experiment over and over, which you can try for yourself by repeatedly flipping a coin, you find she wakes up a third of the time Monday heads, a third of the time Monday tails, and a third of the time Tuesday tails, not 50-25-25 like the previous analysis would suggest.

So if you were Sleeping Beauty and you were awakened and asked "What's the probability the coin came up heads?" What would you say? If you would say one-third, then hit the like button. If you would say one half, hit the dislike button. The answer may seem obvious to you, but you should know that to other people the other answer seems equally obvious. And that's why hundreds and hundreds of philosophy papers have been published on this problem over the past 22 years.

There have been many variations of this problem, like what if instead of being woken up twice if the coin lands tails, she's instead woken up a million times? If the coin comes up heads she's still woken only once. Doesn't it seem absurd in this case, when Sleeping Beauty wakes up, to say that it was just as likely that a coin landed heads as tails? When we know there are a million more wakeups in the tails case than in the heads case.

I mean if you reach into a bag of one white marble and a million black marbles, what are the chances that you pull out that one white marble? I was pretty convinced by this and I considered myself a thirder. But this same argument is used to convince people that we're living in a simulation. The thinking goes that our computing technology has improved so dramatically, even over just the last 40 years, that we can imagine a time in the not-too-distant future when we can create a completely realistic simulation of our world.

And once that occurs, it should be trivial to make unlimited copies of that simulation. And then if you were to ask someone if they're living in a simulation, they would have to admit that they probably are because there are many more instances of that existence than the one true external reality. But how do we know that this hasn't happened already and that we're living inside a simulation? I mean if it can happen then it probably has happened and we are living in a simulation.

This seems like the logical conclusion of the thirder worldview. Now I personally don't buy that I'm living in a simulation, and I think most people don't buy it. But maybe that's just illogical bias. But there's another thought experiment that makes me seriously reconsider the thirder position.

Let's say there's a soccer game between a really great team like Brazil and a less world-dominating team like Canada. So the odds are 80:20 in Brazil's favor. Now a researcher is going to put you to sleep before the game starts. And if Brazil wins, they'll wake you up one time. But if Canada wins, they'll wake you up 30 times in a row, and just like Sleeping Beauty you won't remember if you've been woken before.

Okay, so the game is about to start, you fall asleep... and now you're woken up. Who do you think won the game? The thirder would say Canada, but I would almost certainly say Brazil. I mean why should I give any weight to what the researcher would have done if Canada had won when I'm fairly confident that they won't?

To extend this, let's say Brazil plays Canada five times and we do this experiment each time. Well then if you say Brazil each time you're woken up, you'll probably be right about four out of five of the games. But if you said Canada every time, you would be wrong about those four games but right 30 times in a row when you're repeatedly asked about Canada's one victory.

If you stand to win a bet by correctly answering the question, then by all means you should bet on Canada. But if you want to correctly pick the winner of more of the games, well then you should say Brazil. And this is what's at the heart of the dispute between Halfers and Thirders in the Sleeping Beauty problem. If you want to be right about the outcome of the coin tosses, well you should say the probability of heads is a half, but if you want to answer more questionings correctly, well then you should say one-third.

I want to leave you with one last thought experiment. Imagine that you know for a fact that before our universe began there was a coin flip, and if it came up heads only a single universe would be created. But if it came up tails, a quasi-infinite multiverse would be created and in each of those multiverse universes you'd find every possible variation of Earth and the people on it. In some versions there would be no Earth.

Now you becoming conscious is just like Sleeping Beauty waking up. There's no way to tell if you're in that single universe or in one of the multiverse universes, but you know there are a lot more of them. So would you think that you're for sure in the multiverse? Or are the chances 50/50?

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