> You can’t extract a (meaningful) probability from a single sample no matter how intuitive it seems.

It's possible to get insights from a single sample. Science journals, especially in fields like medicine, do it to talk about atypical case studies. It's a good learning experience, too, when reflecting on the structure and context of those edge cases.

The terms “atypical” and “edge case” are not meaningful with a single sample size that is also the only known observation. There is only the one sample and that’s all you can observe. There’s nothing to compare it against to give “atypical” and “edge case” meaning. Individual samples can be meaningful in a larger distribution, but you still need multiple samples to establish any distribution at all. I think you are referring to outliers within the distribution.

People calculate probabilities with a sample size of zero, even.

For instance, people frequently talk about the probability a given voter will cast the deciding vote in an election.

But we use proxy effects like polls to simulate the actual event. On this case we haven't even been able to do that.