Calling all statistics majors!

So what I’d like to know is whether anyone knows a formula you could use to plug in the levels of your kingdoms (e.g., I have 11 10s and 9 7s at this point), and figure out your odds of 1 tribute, 2 tributes, and so on. I’m competent enough at math to use any formula you give me, but I never took statistics and I can’t find anything online that matches the type of calculation we have to do in this game.

This kind of fits in my obsession of how to best build resources over time, because what I’m basically getting at is whether it makes more sense to level many kingdoms one level at a time, or get one to level 10 before leveling another, *leaving aside the issue of kingdom bonuses, of course.* My instinct tells me it’s better to level multiple kingdoms to have better odds of multiple tributes (and thus keys and gems), but I’m not sure. Obviously if you just want money, you might as well cap out Khaziel, or level Khetar for souls, etc., but I’m really curious about the other side of the coin.