Facetious as it sounds, DG is pretty much spot on.
If there are m = 260 Kingdom cards in total, and you choose n = 10 of them to form a Kingdom, then the probability that none of k are selected is C(m - k, n) / C(m, n), and so the probability that at least one is selected is 1 - C(m - k, n) / C(m, n).
For example, there are 3 Gathering cards, so the chance that none at least one is included in a Kingdom is 1 - C(257, 10) / C(260, 10) = 11.1% or so.
Unfortunately, that calculation is a bit harder if you also want to consider Events and Landmarks, and even moreso if you want to look at intersections of two categories. At that point, you're possibly better off programming a really simplistic Kingdom selector and running it a million times to get a Monte Carlo estimate.