"I can't think of a mathematician who would base a proof on the idea that a number is probably prime."
I can assure you that such a proof probably exists :) Just look at https://en.wikipedia.org/wiki/Probable_prime
Probability theory can be an extremely powerful tool when researching things that are otherwise difficult to reason about. And the theorem statement does not have to be probabilistic for the probabilistic method to be applicable. Just see
As for the following:
"I can't think of an airline passenger who would be totally fine with the flight computer usually being pretty good."
Actually, I would think it's pretty much the opposite. That is, the only type of airline passenger I can think of, is one who is fine with the flight computer (and the airplane in general) usually being pretty reliable. We already know that computers can malfunction and airplanes can crash. Now, of course, how reliable you want the airplane to be is up to you, but if you want it to be flawless, then you should never board an airplane.