I thought about some options a while ago. I would like to see more options for having Colonies and Shelters in games, and I kind of like my third option the best - after drawing your kingdom, flip one card from the kingdom and one from the unused randomisers and if either is from Prosperity then you add Colonies; then do the same to check for Shelters.
TL;DR:
Neirai, who admits to not being able to do probability maths, challenges Method 1 on the basis of the number of total possible Kingdom cards, runs the numbers, and finds that he's more or less wrong although he is hoping for a method that gives approximately 2x higher chances of inclusion.So there's a problem with this method... and I'm going to try to math it out for you.
Disclaimer: Probability math is one of the few maths I can't seem to get my head around. I'm using online calculators to try do the probabilities but while I'm confident enough to say that my errors aren't going to be so huge that you throw out my argument altogether but.. there's going to be some math errors. Probably take everything I say at +or- 10%.
Okay, so right now in the game there's 25 Prosperity Kingdom cards and 314 Kingdom card options (including Prosperity and Renaissance and just in case I'm confusing, this does not include any fan cards at all)
This gives a chance of drawing a single Prosperity randomizer from a stack of all of them of 7.92%
The chances of getting at least 1 Prosperity card in a 10-card Kingdom is slightly less than 37.7%. I say slightly less because you're actually lowering the chance of a subsequent card by 0.03% every time you add a Prosperity card to the Kingdom (removing it from the pool) and increasing it by approximately the same amount because you're also now dealing with a pool that's one card smaller, and you increase it by about 0.03% whenever you put a non-Prosperity card into the pool because those randomizers are gone and this is why this makes my head hurt.
If you use the standard Dominion rule for colonies, you have basically a 10% chance per Prosperity card, so lets say that the chance of doing it with the base set is 3.77% -- 10% of the slightly incorrect number that I said was the chances of getting a set with at least one Prosperity card in it. Which is disgustingly small btw.
Using your method 1, if I'm understanding it correctly, you would be adding an additional step of looking at one of the remaining 304 kingdoms to check if it was a Prosperity card. If you find one, you have a 100% chance of adding Colonies. Let's say for the sake of argument that there's 1 Prosperity card in our kingdom, and 24 left in the randomizers. This gives us an 7.89% chance of getting another Prosperity card. Mashing the two events together in an online calculator, I get... 11.4% chance of either happening.
Now, actually, I started this by saying there's a problem with this. There isn't really: It gives you an 11.4% chance of either happening in the scenario laid out before. If you had 2 Kingdom cards from Prosperity you get a 14.4% chance. If you pulled 0 Kingdom cards you get and 8.22% chance of getting Colonies anyhow.
I tend to write as I think, and not go back, so I said "there's a problem" but there really isn't. If I implement this, I might do a second pull from both pools to inflate the numbers (you get a 15.2% with 0 kingdoms, etc, I'll have to try it out (running out of time aaaaaaaaa)).
It's definitely a nicer method if you're using physical cards, though.