Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice during the experiment, Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening.
A fair coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday. In either case, she will be awakened on Wednesday without interview and the experiment ends.
Any time Sleeping Beauty is awakened and interviewed, she is asked, 'What is your belief now for the proposition that the coin landed heads?
Now if the coin is fair then P(Heads and Monday) = P(Tails and Tuesday)
Therefore P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday).
It must add up to one, thus the 1/3. — Jeremiah
P (Tails and Monday) and P (Heads and Monday) are mutually exclusive — T Clark
Also, P (Tails and Monday) and P (Tails and Tuesday) are not independent. — T Clark
Also, you left out P (Heads and Not Tuesday). — T Clark
They are the same flip. It is pointless to argue independence. That is like saying I rolled a die and got 4 and it is not independent because I got 4. If the coin is tails she will be awakened on Monday and awakened on Tuesday, therefore they have the same probability. — Jeremiah
The trouble is that we cannot use conditional probabilities.So we want the conditional P(H|A). — Srap Tasmaner
P (Tails and Monday) and P (Tails and Tuesday) are the same flip, Are you really suggesting they have a different chance of occurring when they both occur on the same chance event? — Jeremiah
The possible outcomes:
- - M T
H: A, S
T: A, A
If she is awakened before Wednesday, there is one awake on heads and two awake on tails. Of the possible awakens there is a 1/3 chance it is Monday and heads. — Jeremiah
But now I don't know if that's misdirection too. — Srap Tasmaner
And this is consistent with the fact that we know that, given a fair coin toss, there's a 50% chance that it landed heads. We shouldn't change our view of that just because we might be woken up twice rather than once. — Michael
The trouble is that we cannot use conditional probabilities. — andrewk
Mon Tue Heads A S Tails A A
I only presented this side because they led with the 1/2 argument; however, they are correct in pointing out Beauty has gained no additional information. Really all she knows is what she was told before the experiment. — Jeremiah
Yes, Beauty is aware the coin is fair — Jeremiah
but she has also been told the details of the experiment and knows there are three possible events in which she is awakened. If awaken on Tuesday she would not know it is Tuesday; she only knows there are three possible outcomes in which she will be awakened. To Beauty, who does not know if it is Tuesday or Monday when she is awaken, Tails and Tuesdays and Tails and Monday are both valid outcomes. Beauty has to consider three possibilities and only one of them is the desired outcome. — Jeremiah
That makes sense to me. Am I missing something? — Srap Tasmaner
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