The intuition behind the math: If the LHC would not destroy the world, then on date X, a very small number of Everett branches of Earth have the LHC non-working due to a string of random failures, and most Everett branches have the LHC happily chugging ahead. If the LHC would destroy the world, a very small number of Everett branches of Earth have the LHC non-working due to a string of random failures—and most Everett branches have Earth munched up into a black hole.
The very small number of Everett branches that have the LHC non-working due to a string of random failures is the same in both cases.
Thus, if all you know is that you are in an Everett branch in which the LHC is non-working due to a string of random failures, you have no information about whether the other Everett branches have the LHC happily chugging ahead, or dead.
The intuition behind the math: If the LHC would not destroy the world, then on date X, a very small number of Everett branches of Earth have the LHC non-working due to a string of random failures, and most Everett branches have the LHC happily chugging ahead. If the LHC would destroy the world, a very small number of Everett branches of Earth have the LHC non-working due to a string of random failures—and most Everett branches have Earth munched up into a black hole.
The very small number of Everett branches that have the LHC non-working due to a string of random failures is the same in both cases.
Thus, if all you know is that you are in an Everett branch in which the LHC is non-working due to a string of random failures, you have no information about whether the other Everett branches have the LHC happily chugging ahead, or dead.