I'm pretty sure that given the time to learn the calibration, you could make a million largely independent true predictions with a single error, and that having done so the unlikely statements would be less silly than "the LHC will destroy the world". Of course, "independent" is a weasel word. Almost any true observations won't be independent in some sense.

If the failure probability had a known 50% probability of occurring from natural causes, like a quantum coin or some such... then I suspect that if I actually saw that coin come up heads 20 times in a row, I would feel a strong impulse to bet on it coming up heads the next time around."

I would feel such an urge but would override it just as I override the urge to see supernatural forces or the dark lords of the matrix in varied coincidences with similarly low individual probabilities. (I once rolled a die six 8 times out of nine, probability about 2 million. I once lost with a full house in poker, probability a bit under a hundred thousand to one)