simon, that’s right, of course. The reason I’m dragging branches into it is that for the (strong) anthropic principle to apply, we would need some kind of branching—but in this case, the principle doesn’t apply [unless you and I are both wrong], and the math works the same with or without branching.
Eliezer, huh? Surely if F ⇒ S, then F is the same event as (F /\ S). So P(X | F) = P(X | F, S). Unless P(X | F, S) means something different from P(X | F and S)?
Allan, you are right that if the LHC would destroy the world, and you’re a surviving observer, you will find yourself in a branch where LHC has failed, and that if the LHC would not destroy the world and you’re a surviving observer, this is much less likely. But contrary to mostly everybody’s naive intuition, it doesn’t follow that if you’re a suriving observer, LHC has probably failed.
Suppose that out of 1000 women who participate in routine screening, 10 have breast cancer. Suppose that out of 10 women who have breast cancer, 9 have positive mammographies. Suppose that out of 990 women who do not have breast cancer, 81 have a positive mammography.
If you do have breast cancer, getting a positive mammography isn’t very surprising (90% probability). If you do not have breast cancer, getting a positive mammography is quite surprising (less than 10% probability).
But suppose that all you know is that you’ve got a positive mammography. Should you assume that you have breast cancer? Well, out of 90 women who get a positive mammography, 9 have breast cancer (10%). 81 do not have breast cancer (90%). So after getting a positive mammography, the probability that you have breast cancer is 10%...
simon, that’s right, of course. The reason I’m dragging branches into it is that for the (strong) anthropic principle to apply, we would need some kind of branching—but in this case, the principle doesn’t apply [unless you and I are both wrong], and the math works the same with or without branching.
Eliezer, huh? Surely if F ⇒ S, then F is the same event as (F /\ S). So P(X | F) = P(X | F, S). Unless P(X | F, S) means something different from P(X | F and S)?
Allan, you are right that if the LHC would destroy the world, and you’re a surviving observer, you will find yourself in a branch where LHC has failed, and that if the LHC would not destroy the world and you’re a surviving observer, this is much less likely. But contrary to mostly everybody’s naive intuition, it doesn’t follow that if you’re a suriving observer, LHC has probably failed.
Suppose that out of 1000 women who participate in routine screening, 10 have breast cancer. Suppose that out of 10 women who have breast cancer, 9 have positive mammographies. Suppose that out of 990 women who do not have breast cancer, 81 have a positive mammography.
If you do have breast cancer, getting a positive mammography isn’t very surprising (90% probability). If you do not have breast cancer, getting a positive mammography is quite surprising (less than 10% probability).
But suppose that all you know is that you’ve got a positive mammography. Should you assume that you have breast cancer? Well, out of 90 women who get a positive mammography, 9 have breast cancer (10%). 81 do not have breast cancer (90%). So after getting a positive mammography, the probability that you have breast cancer is 10%...
...which is the same as before taking the test.