Manon Bischoff writes
Sleeping Beauty agrees to participate in an experiment. On Sunday she is given a sleeping pill and falls asleep. One of the experimenters then tosses a coin. If “heads” comes up, the scientists awaken Sleeping Beauty on Monday. Afterward, they administer another sleeping pill.
So she sleeps through to Wednesday?
If “tails” comes up, they wake Sleeping Beauty up on Monday, put her back to sleep and wake her up again on Tuesday. Then they give her another sleeping pill. In both cases, they wake her up again on Wednesday, and the experiment ends.
The important thing here is that because of the sleeping drug, Sleeping Beauty has no memory of whether she was woken up before.
In this case, she may have already been the subject of this experiment countless times.
So when she wakes up, she cannot distinguish whether it is Monday or Tuesday. The experimenters do not tell Sleeping Beauty either the outcome of the coin toss nor the day.
Why believe a word which comes out of the mouth of guys who keep drugging you? It is obvious that they have been raping your for years.
They ask her one question after each time she awakens, however: What is the probability that the coin shows heads?
The probability of any coin showing heads is half. It is a different matter that if it was tails, she won't know if it is Tuesday or Wednesday
Depending on the outcome of a coin toss, scientists will wake up Sleeping Beauty either once (heads) or twice (tails).
Why should Sleeping Beauty believe that?
Put yourself in the position of Sleeping Beauty: You wake up, you don’t know what day it is, and you don’t know if you have been woken up before. You only know the theoretical course of the experiment.
My first intuition was that Sleeping Beauty should guess ½.The probability of the coin landing on heads or tails—regardless of the rest of the experiment—is always 50 percent. U.S. philosopher David Lewis held the same view when he learned of the problem. After all, one could even flip the coin before sending Sleeping Beauty to sleep. By the experiment’s design, she does not have any extra clues to the situation, so logically she should state the probability as ½.
But there are also conclusive arguments in favor of a probability of ⅓. If you think through Sleeping Beauty’s experience, then three scenarios can occur:
She wakes up Monday, and heads was thrown.
She wakes up Monday, and tails was thrown.
She wakes up Tuesday, and tails was thrown.
There are many more scenarios. No coin was thrown. There was never any sleeping pill. The experimenters are lying or the got mixed up and dosed themselves etc. etc.
In this case even if the experimenters have to tell the truth they would not be able to do so.
What are the probabilities for each event? You can investigate this both mathematically and empirically. Suppose you flip a coin 100 times and get tails 52 times and heads 48 times. Put another way, the Monday/heads scenario occurs 48 times, and Monday/tails and Tuesday/tails occur 52 times each.
Because Tuesday/tails always follows Monday/tails, the probabilities for all three events are equal—and must therefore be ⅓.
Why stop there? Why not say Tuesday plus feeling your period is about to start and Wednesday and your period actually having started. This extra information changes the probability.
When Sleeping Beauty is awakened and asked to answer what the probability of the coin toss was for heads, she should therefore answer ⅓, according to this reasoning.
She can answer in any way she likes depending on period pains etc.
If after 100 flips, you got heads 48 times and tails 52 times, you could apply those numbers to Sleeping Beauty’s Monday and Tuesday scenarios. You’d discover that these three situations are more or less equally likely to occur.
Philosopher of science Adam Elga of Princeton University, who popularized the Sleeping Beauty problem in 2000, came to this conclusion. He formulated his argument in a mathematically sound way. If Sleeping Beauty is told when she wakes up that today is Monday (M), then the probability of Monday/heads (M, H) and Monday/tails (M, Z) is indisputably equal: P(M, H) = P(M, Z) = ½, where P stands for probability. On the other hand, if Sleeping Beauty wakes up and learns that tails have been thrown, then that day could equally be either Monday or Tuesday (T), meaning P(M, Z) = P(T, Z) = ½.
Actually, if you are dealing with experiments involving memory erasing drugs, anything at all is possible. You may be a secret agent with implanted memories. The experimenters may be trying to destroy those implanted memories so that you can be tortured into giving up various secrets.
There are an infinite number of possibilities. Absent some 'biological clock' based information- e.g. period pains- the maximal uncertainty principle applies. So the answer is half.
This puzzle has some interesting applications. Philosophers and mathematicians can use it to think about decision-making and probability broadly. For example, this thought experiment illustrates how someone’s beliefs—in this case, Sleeping Beauty’s—can lead to more than one rational conclusion.
I think it suggests that additional information changes probability even if that information is not certain.
It also underscores the difference between the number of experimental possibilities (such as flipping heads versus tails) and the possible experiences of someone within an experiment.
I think there is a connection between Elga's Sleeping Beauty & Watanabe's Ugly Duckling theorem which suggests that you can't get rid of all bias. Essentially, to get away from 'maximal uncertainty' some arbitrary assumptions or mechanisms are necessary. On the other hand, both may be fairy stories because though we may have a model for rational choice we don't know how it could itself be rationally chosen.
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