But it’s still wrong to assign them 50/50 odds.
Do not hit the like or dislike buttons yet. I want you to consider a problem that has been one of the most controversial in math and philosophy over the past 20 years, for which there is no consensus answer.
Listen to the problem and then vote for the answer you prefer using the like and dislike buttons.
The setup is as follows: Sleeping Beauty volunteers to be the subject of an experiment. Before it starts, she is informed of the procedure. On Sunday night, she will be put to sleep and a fair coin will be flipped. If the coin comes up heads, she will 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. Each time she gets put back to sleep, she will forget that she was ever awakened. In the brief period anytime she is awake, she will be told no information. But she will be asked one question: “What do you believe is the probability that the coin came up heads?”
So how should she answer?
The intuitive answer that pops into one’s 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.
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.
Others would argue that something does change when she is awakened. She actually learns something important - she learns that she has gone from existing in a reality where there are two possible states (heads or tails) to existing in a reality where there are three possible states (Monday heads, Monday tails or Tuesday tails). 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.
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 would be wrong to assign them 50/50 odds, as 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, but it is still wrong to assign them 50/50 odds. The implied question is: “Given you’re awake, what’s the probability that the coin came up heads?” And that is one-third. 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, thus the tails probability gets split across two days, 25 percent each. However, if you repeat the experiment over and over, you find that Sleeping Beauty 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.
If you were Sleeping Beauty and were asked “What’s the probability the coin came up heads?”, the answer may seem obvious to some, but hundreds of philosophy papers have been published on this problem. There have been many variations of this problem, such as what if instead of being woken up twice if the coin lands tails, she’s instead woken up a million times?
This same argument is used to convince people that we’re living in a simulation, as our computing technology has improved so dramatically. There is also another thought experiment that makes one seriously reconsider the third or position - if there’s a soccer game between a great team like Brazil and a less World dominating team like Canada, and the researcher puts you to sleep before the game starts, if Brazil wins they’ll wake you up one time, but if Canada wins they’ll wake you up 30 times in a row.
If you stand to win a bet by correctly answering the question then by all means you should bet on Canada. If you want to correctly pick the winner of more of the games, you should say Brazil. This is 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, you should say the probability of heads is a half, but if you want to answer more questions correctly, you should say one-third.
To develop intuition about probability, one should work through scenarios or run simulations like we did for Sleeping Beauty. This video’s sponsor, Brilliant, offers probability courses that walk through situational probability questions. You can try it free for 30 days at Brilliant.org/Veritasium. With Brilliant, you can model the real world with math and expand your understanding of cutting edge topics like AI and machine learning. If you want to get started with Brilliant’s annual premium subscription, they are giving 20 percent off to the first 200 people to sign up via my link.