Actually, failures of the LHC should never have any effect at all on our estimate of the probability that if it did not fail it would destroy Earth.
This is because the ex ante probability of failure of the LHC is independent of whether or not if it turned on it would destroy Earth. A simple application of Bayes’ rule.
Now, the reason you come to a wrong conclusion is not because you wrongly applied the anthropic principle, but because you failed to apply it (or applied it selectively). You realized that the probability of failure given survival is higher under the hypothesis that the LHC would destroy the Earth if it did not fail, but you didn’t take into account the fact that the probability of survival is itself lower under that hypothesis (i.e. the anthropic principle).
Actually, failures of the LHC should never have any effect at all on our estimate of the probability that if it did not fail it would destroy Earth.
This is because the ex ante probability of failure of the LHC is independent of whether or not if it turned on it would destroy Earth. A simple application of Bayes’ rule.
Now, the reason you come to a wrong conclusion is not because you wrongly applied the anthropic principle, but because you failed to apply it (or applied it selectively). You realized that the probability of failure given survival is higher under the hypothesis that the LHC would destroy the Earth if it did not fail, but you didn’t take into account the fact that the probability of survival is itself lower under that hypothesis (i.e. the anthropic principle).