Yes, both of these credences should obey the axioms of a probability space.
This sort of thing is applied in cryptography with the concept of “probable primes”, which are numbers (typically with many thousands of decimal digits) that pass a number of randomized tests. The exact nature of the tests isn’t particularly important, but the idea is that for every composite number, most (at least 3⁄4) of the numbers less than it are “witnesses” such that when you apply a particular procedure using that number, the composite number fails the test but primes have no such failures.
So the idea is that you pick many random numbers, and each pass gives you more confidence that the number is actually prime. The probability of any composite number passing (say) 50 such tests is no more than 4^-50, and for most composite numbers it is very much less than that.
No such randomized test is known for parity of the googolth digit of pi, but we also don’t know that there isn’t one. If there was one, it would make sense to update credence using the results of such tests using probability axioms.
Yes, both of these credences should obey the axioms of a probability space.
This sort of thing is applied in cryptography with the concept of “probable primes”, which are numbers (typically with many thousands of decimal digits) that pass a number of randomized tests. The exact nature of the tests isn’t particularly important, but the idea is that for every composite number, most (at least 3⁄4) of the numbers less than it are “witnesses” such that when you apply a particular procedure using that number, the composite number fails the test but primes have no such failures.
So the idea is that you pick many random numbers, and each pass gives you more confidence that the number is actually prime. The probability of any composite number passing (say) 50 such tests is no more than 4^-50, and for most composite numbers it is very much less than that.
No such randomized test is known for parity of the googolth digit of pi, but we also don’t know that there isn’t one. If there was one, it would make sense to update credence using the results of such tests using probability axioms.