But the thirder position would say that it’s still one-third because she’s only asked to consider the probability given that she’s awake.
Do not hit the like or dislike button yet! I want you to consider a very controversial problem that has been debated in math and philosophy for the past 20 years without any 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.
Here is the setup: 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. 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 asked one question: What do you believe is the probability that the coin came up heads?
So, how should she answer? Halfers would say that the probability of a coin coming up heads is one-half 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.
Thirders would argue that something does change when she is awakened. She learns that she goes 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, making the probability that the coin came up heads one-third.
Now, which answer do you prefer? You can vote using the like and dislike buttons. 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 with 25 percent each. However, if you repeat the experiment multiple times, you find that Sleeping Beauty wakes up a third of the time with Monday heads, a third of the time with Monday tails, and a third of the time with Tuesday tails, not 50-25-25 like the previous analysis would suggest. 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 same argument is used to convince people that we’re living in a simulation. If the coin comes up heads, Sleeping Beauty is still woken up only once, but if the coin comes up tails, she is woken up a million times. 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?
Another thought experiment to consider is if there’s a soccer game between a great team like Brazil and a less World dominating team like Canada, with the odds being 80:20 in Brazil’s favor. A researcher puts you to sleep before the game starts. If Brazil wins, they’ll wake you up once, but if Canada wins, they’ll wake you up 30 times in a row. If you say Brazil each time you’re woken up, you’ll be right about four out of five of the games. However, 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 you should bet on Canada. 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 answer more questions correctly, you should say the probability of heads is a half, and one-third respectively.
To develop intuition about probability, one should work through scenarios or run simulations, such as the one done for Sleeping Beauty. Brilliant offers probability courses that walk through many situational probability questions and allow you to build your own simulations. With Brilliant, you can model the real world with math and expand your understanding of cutting edge topics like AI and machine learning.
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