We avoid Peirce’s observation? through careful modeling, consistent methods, developed intuition, result analysis, and reasonableness tests. We must also prepare for non-rational probabilities and non-uniform probability distributions.
Uniform probability assignments allow computations to be either the ratio of satisfactory outcomes over all outcomes, or the sum of individual satisfactory outcome probabilities. Non-uniform probability calculations can only be carried out by summing individual probabilities. Where you end up depends on the direction you start walking. “Random” without modifier is the code-word for uniform probability assignments.
We now discover tools for computing probabilities over finite discrete sample spaces, the related uniform solution methods?, and the intuition to guide their use, in part by appealing to the frequency interpretation of probability.