Mathematics usually has a clear answer. Especially if the task is not too complicated. However, there is still no universal consensus on the Sleeping Beauty issue of 2000. Philosophical and mathematicians split into his two camps, citing incessantly (often very convincingly) the arguments of each camp. There are over 100 technical publications about the puzzle, and nearly everyone who has heard of the Sleeping Beauty thought experiment has developed their own strong opinion.
Here are the questions that have plagued the minds of experts: Sleeping Beauty agrees to participate in the experiment. On Sunday she is given sleeping pills and falls asleep. Then one of her experimenters flips a coin. If the “head” appears, the scientist will awaken Sleeping Beauty on Monday. She then administers another sleeping pill. Wake up Sleeping Beauty on Monday when the tail comes out, put it to sleep, wake it up again on tuesday. They then give her another sleeping pill. In either case, wake her up again on Wednesday and her experiment will end.
[Read about the importance of thought experiments to Albert Einstein]
The important thing here is that because of the sleeping pills, Sleeping Beauty has no memory of whether she has woken up before. The experimenter does not tell Sleeping Beauty the result of the coin toss, nor on the day of the experiment.
But they ask her one question every time she wakes up: What is the probability that the coin will show heads?
Put yourself in the position of Sleeping Beauty. When you wake up, you don’t know what day it is, and you don’t know if you’ve woken up before. You only know the theoretical course of the experiment.
My first intuition was that Sleeping Beauty should hit 1/2. Regardless of the rest of the experiment, she always has a 50% chance of the coin landing heads or tails. American philosopher David Lewis expressed the same view when he learned of the problem. After all, you can even flip a coin before putting Sleeping Beauty to sleep. Due to the design of her experiment, she has no extra clues to the situation, so logically we should state the probability as ½.
However, there are also conclusive arguments in favor of the 1/3 probability. Given the Sleeping Beauty experience, we can think of three possible scenarios for her:
She woke up on Monday to find her head thrown.
She woke up on Monday to find her tail thrown.
She woke up on Tuesday to find her tail thrown.
What is the probability of each event? We can find this out both mathematically and empirically. Suppose he flips a coin 100 times and gets 52 tails and 48 heads. In other words, the Monday/Heads scenario occurs 48 times, while Monday/Back and Tuesday/Back occur 52 times each.
Since Tuesday/Tail always follows Monday/Tail, the probabilities of all three events must be equal and therefore 1/3. For this reason, when Sleeping Beauty wakes up and is asked what her coin toss odds are heads, she should answer 1/3 of hers.
Princeton University philosopher of science Adam Elga, who popularized the Sleeping Beauty problem in 2000, came to this conclusion. He formulated his claim in a mathematically correct way. The probability of tails (M, Z) is definitely equal: P(M, H) = P (M, Z) = ½, where P represents the probability. On the other hand, if Sleeping Beauty wakes up and finds out that her tail has been thrown, that day could be either Monday or Tuesday (T), and P(M, Z) = P (T, Z) = ½.
According to the conditional probability calculation, in the general case (Sleeping Beauty receives no additional information) the three values are equal: P(M, Z) = P(M, H) = P(T, Z) ). All three probabilities sum to 1, so the individual values are 1/3. In other words, Sleeping Beauty awakens her tail twice as often as her head, so from Elga’s point of view the answer should be 1/3.
take it to the extreme
Now that you’ve heard the two main arguments, how would you answer this question? To better understand the Sleeping Beauty problem, consider a more extreme version of the thought experiment. prize.
In the case of the tail, suppose Sleeping Beauty wakes up and is interrogated not just once more the next day, but a million times (perhaps at shorter intervals – even for fairy tale characters, this schedule is tight. to become a thing). If you wake her up and ask her the odds of her coin turning heads, the answer ½ doesn’t seem logical in this scenario. If the result of the coin toss is tails, Sleeping Beauty will be asked a million times in a row, and if it is heads, it will be asked only once.
However, in extreme cases, it is possible to strengthen the position of the ½ side. For example, instead of a coin toss, a sports bet such as a foot race pitting retired sprinter Usain Bolt against singer Taylor Swift could be used. In this scenario, as many would expect, if Bolt, the world record holder in his category for multiple runs, beat Pop Her Star, Sleeping Beauty would wake her up only once on Monday. increase. But if, against all odds, Swift proves to be faster, Sleeping Beauty says she has to wake up 30 times in a row every day for a month. The odds of Bolt losing to Swift are very low. But if we apply the same logic that caused 1/3 of the reactions, we should treat these scenarios with equal weight. Sleeping Beauty still has to bet on her Swift wins after she wakes up. Lewis felt the discussion was pointless. This thought experiment, as he argued, ½ faction.
Are you completely confused now? you are not alone. Has your opinion changed? I have Either way, I’m not entirely convinced by her 1/2 side. You can also gain insight from the 1/3 position.
This puzzle has several interesting applications. Philosophers and mathematicians can use it to think broadly about decision-making and probability. For example, this thought experiment shows how someone’s beliefs (in this case, Sleeping Beauty) can lead to multiple rational conclusions. It also highlights the difference between a number of experimental possibilities (such as flipping heads and tails) and someone’s possible experience. internal experiment.
This article originally appeared on science spectrum Reproduced with permission.
