For example, I would be substantially more alarmed about a lottery device with a well-defined chance of 1 in 1,000,000 of destroying the world, than I am about the Large Hadron Collider switched on. If I could prevent only one of these events, I would prevent the lottery.

On the other hand, if you asked me whether I could make one million statements of authority equal to "The Large Hadron Collider will not destroy the world", and be wrong, on average, around once, then I would have to say no.

Hmm... might this be the heuristic that makes people prefer a 1% chance of 1000 deaths to a definite death for 5? The lottery would definately destroy worlds, with as many deaths as killing over six thousand people in each Everett branch. Running the LHC means a higher expected number of dead worlds by your own estimates, but it's all or nothing across universes. It will most probably just be safe.

If you had a definate number for both P(Doomsday Lottery Device Win) and P(Doomsday LHC) you'd shut up and multiply, but you haven't so you don't. But you still should because you're pretty sure P(D-LHC) >> P(DLDW) even if you don't know a figure for P(DLHC).

This assumes Paul's assumption, above.