The problem vexing the minds of experts is as follows: 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. 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.
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As other responses have mentioned you are missing half the question, the part where she doesn’t remember what happened previously when she is put to sleep and she is asked the probability the coin flip was heads. Also like many similar questions this isn’t really an issue of probability but instead revolves around precisely what is being asked. It is semantics, not mathematics.
In essence they have taken Sleeping Beauty and put her in a situation where she is being asked about an event with 50/50 chances (the coin flip), but engineered a situation where she answers more frequently when the coin lands on tails. If she guesses that the coin flip was heads then she would be wrong 2/3 times since she answers twice when it is tails!
This might be confusing, but only for people who are good at probabilities but have poor language skills. Sleeping Beauty is being asked:
> “What is the probability that the coin shows heads?”
She isn’t being asked what she thinks the coin flip was, so answering “50%” when the coin shows tails isn’t wrong. It is correct no matter what the coin flip turned out to be or how many times she is asked!
Now if she were asked something subtly different like “What is the probability you are awakening to a coin flip that landed heads?” then the answer of “1/3” makes sense.
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