To clarify:

When you find a probability through bayes-fu, you need a prior. Precisely one prior. Which is going to be pretty arbitrary.

But what if you picked two symmetrical priors, and applied the evidence to those? You start with prior 1: 99.99% Prior 2: 0.01%

When you have a reasonable amount of evidence you'll find that prior 1 and prior 2 result in similar values (ie. prior 1 might give 40% while prior 2 gives 25%) If you were to get more evidence, the difference would gradually vanish, and the smaller the probability range the more firm your probability is.