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The “Sleeping Beauty Problem” has obviously attracted mixed responses – some people say 1/3 and others say 1/2. I’m having trouble working out which group I agree with. In the meantime, here is another thought experiment designed to help us decide if we trust the SIA or the main alternative, the Self-Sampling Assumption (SSA), suggested by Nick Bostrum.
“II. POSSIBLE VS. EXISTING
A. God’s Coin Toss
The crux of the matter can be described by a “God’s Coin Toss” experiment [3,6]. Suppose that God tosses a fair coin. If it comes up heads, he creates ten people, each in their own room. If tails, he creates one thousand people, each in their own room. The rooms are numbered 1-10 or 1-1000. The people cannot see or communicate with the other rooms. Suppose that you know all this, and you discover that you are in one of the first ten rooms. How should you reason that the coin fell? Leslie and Bostrom argue as follows. Before you look at your room number, you should think that since the coin was fair the chance of heads was 1/2. Now if the coin was heads, then of course you would be in one of the first ten rooms. However, if the coin was tails, the chance to be in one of the first ten rooms is 1/100. Thus, according to Eq. (2), you should now believe that the coin was heads with probability 0.99. The alternative argument runs as follows. Before you look at your room number, you should think that the probability of heads is 0.99. There are one thousand possible people who would be right with that belief, whereas only ten would be right with the belief in heads. When you look at your room number, you should then update your probabilities using Eq. (2). The result is that in the end you think the chance is 1/2 that the coin was heads. Another way to say the same thing is that there are ten ways to have the coin heads and you in a room in the first ten, and ten ways to have the coin tails and you in a room in the first ten, and thus the chances for heads and tails are equal. The difference here hinges on whether one considers possible people in the same ways that one considers actual people. If instead of flipping a coin, God creates both sets of rooms, then Leslie and Bostrom and I all agree that you should think it much more probable that you are in the large set before you look at your room number, and equally probable afterward. Treating the two possibilities in the same way as two sets of actual observers implies the Self-Indication Assumption: the existence of a large number of observers in a possible universe increases the chance to find oneself in that universe.
I will argue below that the equal treatment of possible and actual observers is correct.