## The Sleeping Beauty problem

A recent post by Sean Carroll reignited a debate about the "Sleeping beauty problem".
This is a simple problem of probabilities, involving tossing a coin. But for some reason, it seems to be no agreement about its solution.

Consider the following experiment:
- On Sunday, put Sleeping Beauty to sleep.
- Toss a fair coin.
- We are interested to ask Sleeping Beauty the question
Q. What is your belief now for the proposition that the coin landed heads?
- If the coin comes up heads, wake up Sleeping Beauty on Monday and ask her the question. Then drug her to forget that awakening.
- If the coin comes up tails, wake up Sleeping Beauty both on Monday and Tuesday and ask her the question. Each time drug her to forget that awakening.
- In both cases, don't forget to wake her on Wednesday and end the experiment.

If you have trouble convincing a Beauty to let you put her to sleep and drug her, you can try your luck with people who already have very short memory, like Lucy Whitmore from "50 First Dates", Leonard from "Memento", Allie from The Notebook, or Dory from "Finding Nemo". Or you can make the experiment with Dory from "Finding Nemo".

You can also perform the experiment with Dory from "Finding Nemo".

Those thinking they know the answer are mainly in one of two camps: halfers, who think she should answer 1/2, and thirdirs, who think she should answer 1/3. Thirdirs say that when Beauty is waken and interviewed, she thinks she can be in one of three situations. Since only in one of the cases the coin turned up heads, the answer must be 1/3. Halfers say that this answer is wrong, being probably caused by drug abuse, and since the coin is fair, the answer should be 1/2. There is nothing that can provide new information to Sleeping Beauty, so this answer should remain 1/2.

I will not detail here the debates still ongoing on the net, and the articles which are written about this. I just want to explain why I think that this debate is based on different understandings of the question.

## Another experiment

Consider the following experiment.
- Prepare a large box, in which you can put apples and oranges, without seeing its content.
- Toss a fair coin.
- If the coin comes up heads, put one orange in the box.
- If the coin comes up tails, put two apples in the box.
- Repeat this many times.
- At the end, randomly extract a fruit from the box. Unless the experiment took too long, the fruits are not yet rotten, so you can extract a fruit.
- Then answer the following questions:
1. What is the probability that the fruit you will extract was introduced after the coin landed heads?
2. What fraction of the total times the coin was tossed, it landed heads?

The answer to question 1 is of course 1/3, because 1/3 of the fruits are oranges, and oranges were placed in the box when the coin landed heads.

The answer to question 2 is of course 1/2, because the coin is supposed to be fair.

My claim is that thirdirs were actually answering question 1, and halfers were answering question 2.

The question Sleeping Beauty was asked can be seen as equivalent to both question 1 and question 2.