I think people are just choosing comparison sets wrong? Using the bag anology, there are two pieces of paper labeled “6” so upon being handed a random piece of paper labeled “6″ I should be 50⁄50 as to whether it came from the first bag or the second bag, No?
Imagine someone named Omega offers to play a game with you. Omega has a bag, and they swear on their life that exactly one of the following statements is true:
They put a single piece of paper in the bag, and it has “1” written on it.
They put 10 trillion pieces of paper in the bag, numbered “1”, “2″, “3”, etc. up to ten trillion.
Omega then has an independent neutral third party reach into the bag and pull out a random piece of paper which they then hand to you. You look at the piece of paper and it says “1” on it. Omega doesn’t get to look at the piece of paper, so they don’t know what number you saw on that paper.
Now the game Omega propose to you is: If you can guess which of the two statements was the true one, they’ll give you a million dollars. Otherwise, you get nothing.
Which do you guess? Do you guess that the bag had a single piece of paper in it, or do you guess that the bag had 10 trillion pieces of paper in it?
In order for such an experiment to yield useful data, you would need to repeat it some number of times (the more, the better, of course); and you would then observe whether the ratio of the number of “a 6 was drawn from the smaller bag” outcomes to the number of “a 6 was drawn” outcomes approached 0.5, or almost-1 (~0.9998, I believe it would be?).
However, there are two different ways of repeating the experiment. One way is to do it with replacement, and the other is to do it without replacement.
If you repeat the experiment with replacement, then, of course, the great majority of 6s will have been picked from the smaller bag.
But if you repeat the experiment without replacement, then exactly two 6s will ever be picked; and of those two, one will have been picked from the smaller bag. That gives us a probability of 0.5.
Now, in the one-off (non-repeated) experiment, with the bags of papers, it’s not clear that there’s any meaning to asking whether we should reason analogously to the repeated experiment without replacement, or analogously to the repeated experiment with replacement. (After all, just one outcome ever occurs.)
But in the Doomsday Argument case, there does seem to be an argument to be made that we should reason analogously to the repeated experiment without replacement… after all, presumably two disembodied souls cannot be born as the same particular person, right? Once a given individual-physical-history “slot” is “occupied”, that’s it; it can’t be picked again. (Or so we might intuitively reason.)
Of course, this sort of thing only highlights once again the fundamental absurdity of the model…
I think people are just choosing comparison sets wrong? Using the bag anology, there are two pieces of paper labeled “6” so upon being handed a random piece of paper labeled “6″ I should be 50⁄50 as to whether it came from the first bag or the second bag, No?
Imagine someone named Omega offers to play a game with you. Omega has a bag, and they swear on their life that exactly one of the following statements is true:
They put a single piece of paper in the bag, and it has “1” written on it.
They put 10 trillion pieces of paper in the bag, numbered “1”, “2″, “3”, etc. up to ten trillion.
Omega then has an independent neutral third party reach into the bag and pull out a random piece of paper which they then hand to you. You look at the piece of paper and it says “1” on it. Omega doesn’t get to look at the piece of paper, so they don’t know what number you saw on that paper.
Now the game Omega propose to you is: If you can guess which of the two statements was the true one, they’ll give you a million dollars. Otherwise, you get nothing.
Which do you guess? Do you guess that the bag had a single piece of paper in it, or do you guess that the bag had 10 trillion pieces of paper in it?
Eh I didn’t think you can just ignore facts like the components of the bag. You could actually do this experiment, and the probability won’t be 50%.
In order for such an experiment to yield useful data, you would need to repeat it some number of times (the more, the better, of course); and you would then observe whether the ratio of the number of “a 6 was drawn from the smaller bag” outcomes to the number of “a 6 was drawn” outcomes approached 0.5, or almost-1 (~0.9998, I believe it would be?).
However, there are two different ways of repeating the experiment. One way is to do it with replacement, and the other is to do it without replacement.
If you repeat the experiment with replacement, then, of course, the great majority of 6s will have been picked from the smaller bag.
But if you repeat the experiment without replacement, then exactly two 6s will ever be picked; and of those two, one will have been picked from the smaller bag. That gives us a probability of 0.5.
Now, in the one-off (non-repeated) experiment, with the bags of papers, it’s not clear that there’s any meaning to asking whether we should reason analogously to the repeated experiment without replacement, or analogously to the repeated experiment with replacement. (After all, just one outcome ever occurs.)
But in the Doomsday Argument case, there does seem to be an argument to be made that we should reason analogously to the repeated experiment without replacement… after all, presumably two disembodied souls cannot be born as the same particular person, right? Once a given individual-physical-history “slot” is “occupied”, that’s it; it can’t be picked again. (Or so we might intuitively reason.)
Of course, this sort of thing only highlights once again the fundamental absurdity of the model…