Those probabilities are mostly pulled out of thin air, but I've seen other people use them, so apparently it's expected. Is there some systematic method people use to arrive at them, or do you just sort of look out a window and see what number feels right?

The probability you assign to a hypothesis should accurately represent your degree of belief that the hypothesis is true. Moreover, your degree of belief should be coherent with the rules of probability theory. Unfortunately, we human beings are notoriously bad at probabilistic reasoning. So while there are systematic methods for assigning probabilities based on evidence, it takes a lot of work to use them properly. For a lot of untrained people, myself included, the best we can currently do is see how we feel, attempt to quantify it, and try to constrain it based on rational factors.

If you want to learn more, a few key search words here are "Bayes' theorem," "heuristics and biases," and "debiasing." If you read through the sequences - a daunting task, I know - a lot of it is covered in detail. Or if you'd prefer to read some academic papers and books on the subject, I'm sure I and other users could make recommendations.