Nice example of how using a probability of exactly zero can screw you over. Two observations.

1. Could have done with a link to Eliezer's 0 and 1 are not probabilities from back in 2008.

2. I say you can get to 99.99% confidence that 1159 is prime (if it actually is; I haven't checked); probably 99.9999%. Suppose you (a) write a program to check all possible divisors, test that it gives the right answers for everything up to 100, and run it multiple times in case of cosmic rays; (b) look it up in a table of prime numbers; (c) apply, by hand, one of the fancy number-theoretical primality tests (most of these are probabilistic -- but again you can find statements of the form "If a number less than 10^12 passes these specific tests and isn't one of the following short list of exceptions, it is prime"). Then I reckon that apart from theories where what's wrong is your brain a,b,c are clearly extremely close to independent; (a) has well less than 0.001 chance of failure, (b) well less than 0.01, and (c) well less than 0.1; so the only hypothesis you need to consider that might take the probability above 10^-6 is that you're delusional in some way that specifically messes up your ability to tell whether 1159 is prime. Now (d) this is surely extraordinarily rare -- delusions in general aren't so rare, but this is something very specific and weird; and (e) if your mind is that badly screwed up then attempting to work with probabilities is probably kinda meaningless anyway. (Theorems like Cox's say that you should use probabilities as measures of confidence, and compute with them in the standard ways, if you are a rational agent. If you are known to be a catastrophically irrational agent, then what probabilities you assign is probably not the greatest of your worries.)