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…
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…