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.
Recently the Large Hadron Collider was damaged by a mechanical failure. This requires the collider to be warmed up, repaired, and then cooled down again, so we're looking at a two-month delay.
Inevitably, many commenters said, "Anthropic principle! If the LHC had worked, it would have produced a black hole or strangelet or vacuum failure, and we wouldn't be here!"
This remark may be somewhat premature, since I don't think we're yet at the point in time when the LHC would have started producing collisions if not for this malfunction. However, a few weeks(?) from now, the "Anthropic!" hypothesis will start to make sense, assuming it can make sense at all. (Does this mean we can foresee executing a future probability update, but can't go ahead and update now?)
As you know, I don't spend much time worrying about the Large Hadron Collider when I've got much larger existential-risk-fish to fry. However, there's an exercise in probability theory (which I first picked up from E.T. Jaynes) along the lines of, "How many times does a coin have to come up heads before you believe the coin is fixed?" This tells you how low your prior probability is for the hypothesis. If a coin comes up heads only twice, that's definitely not a good reason to believe it's fixed, unless you already suspected from the beginning. But if it comes up heads 100 times, it's taking you too long to notice.
So - taking into account the previous cancellation of the Superconducting Supercollider (SSC) - how many times does the LHC have to fail before you'll start considering an anthropic explanation? 10? 20? 50?
After observing empirically that the LHC had failed 100 times in a row, would you endorse a policy of keeping the LHC powered up, but trying to fire it again only in the event of, say, nuclear terrorism or a global economic crash?