So why can’t we just assign a 50/50 probability? Well the thirder position would argue that the question is not about the coin it’s about the awakening and the fact that she learns something when she wakes up.
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 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 why can’t we just assign a 50/50 probability? Well the thirder position would argue that the question is not about the coin it’s about the awakening and the fact that she learns something when she wakes up. But if you’re just trying to figure out what’s true then it seems like you should go with Brazil.
The chance of waking up on Monday with heads is 50 percent and the chance of waking up on Monday or Tuesday with tails is 50 percent, so the tails probability gets split across two days 25 percent each. However, if you repeat the experiment over and over, you find that you wake 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 as the previous analysis suggested.
If you were Sleeping Beauty and you were asked “What’s the probability the coin came up heads?”, the answer may seem obvious to some, but to others the other answer seems equally obvious. This is why hundreds of philosophy papers have been published on this problem over the past 22 years.
Another thought experiment is if Brazil were to play Canada five times and a researcher puts you to sleep before the game starts. If Brazil wins, you are awoken once, and if Canada wins, you are awoken 30 times in a row. If you say Brazil each time you’re woken up, you’ll probably be right about four out of five of the games. However, if you said Canada each time, you would be wrong about those four games, but right 30 times in a row when asked about Canada’s one victory. If you stand to win a bet by correctly answering the question, then you should bet on Canada. But if you’re just trying to figure out what’s true, it seems like you should go with Brazil. If you want to correctly pick the winner of more of the games, then you should say Brazil. This is the heart of the dispute between Halfers and Thirders in the Sleeping Beauty problem. To be right about the outcome of the coin tosses, you should say the probability of heads is a half, but if you want to answer more questions correctly, you should say one-third.
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. In each of those Multiverse universes, you’d find every possible variation of Earth and the people on it. So, would you think that you are for sure in the Multiverse, or are the chances 50/50?
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