The Sleeping Beauty Problem (SBP) is a puzzle in probability theory and on self-locating belief.
From the wiki on
The Sleeping Beauty Problem
The paradox imagines that Sleeping Beauty volunteers to undergo the following experiment. On Sunday she is given a drug that sends her to sleep. A fair coin is then tossed just once in the course of the experiment to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday, given a second dose of the sleeping drug, and awakened and interviewed again on Tuesday. The experiment then ends on Tuesday, without flipping the coin again. The sleeping drug induces a mild amnesia, so that she cannot remember any previous awakenings during the course of the experiment (if any). During the experiment, she has no access to anything that would give a clue as to the day of the week. However, she knows all the details of the experiment.
Each interview consists of one question, “What is your credence now for the proposition that our coin landed heads?”
This problem is considered paradoxical because the answer is often given as either 1/3 or 1/2.
It may appear that the different probabilities, as determined by Beauty and the experimenter, are due to their different levels of knowledge. This is not so. Beauty’s amnesia and ignorance of the day of the week are irrelevant. Even her waking up is irrelevant. The only factor that is relevant to the different probability calculations is sampling. If Beauty knew whether it was Monday or Tuesday then she would give the following odds. If it were Monday then the odds would be 1/2. If Tuesday then the odds would be 100% tails. Putting the two cases together we still get 1/3 chance of heads.
Which is the correct solution?
In this paper by Nick Bostrom , a hybrid solution is proposed.
From the abstract:
Opinion on this problem is split between two camps, those who defend the “1/2 view” and those who advocate the “1/3 view”. I argue that both these positions are mistaken. Instead, I propose a new “hybrid” model, which avoids the faults of the standard views while retaining their attractive properties.
The Self-Indication Assumption:
That assumption states that each observer should regard her own existence as evidence supporting hypotheses that imply the existence of a greater total population of observers in the world, the degree of support being proportional to the implied (expected) number of observers.
And the presumptuous philosopher:
It is the year 2100 and physicists have narrowed down the search for a theory of everything to only two remaining plausible candidate theories, T1 and T2 (using considerations from super-duper symmetry). According to T1 the world is very, very big but finite and there are a total of a trillion trillion observers in the cosmos. According to T2, the world is very, very, very big but finite and there are a trillion trillion trillion observers. The super-duper symmetry considerations are indifferent between these two theories. Physicists are preparing a simple experiment that will falsify one of the theories. Enter the presumptuous philosopher: “Hey guys, it is completely unnecessary for you to do the experiment, because I can already show to you that T2 is about a trillion times more likely to be true than T1!” (Whereupon the presumptuous philosopher explains the Self-Indication Assumption.)
Predicament in cosmology:
It is worth noting that the situation described in this modified version of Presumptuous Philosopher is by no means a farfetched possibility. Contemporary cosmologists face essentially that predicament. They are trying to determine whether the universe is finite or infinite.
If the universe is infinite then with probability one there are an infinite number of agent-moments in states subjectively indistinguishable to your current one. Therefore, if Elga’s 1/3 view is correct, we could conclude that we already have “infinitely strong” evidence that the universe is infinite.