Very clever! It is indeed true that if you forget all previous marble paintings, the best way to ensure that 2/3 get painted one color is to paint it that color with p = 2/3.

And interestingly, I can think of several examples of my own life when I've been in that situation. For example, when I'm playing Alpha Centauri, I want to make sure I have a good mix of artillery, infantry, and speeders, but it's tedious to keep track of how many I have of each, so I just pick in a roughly random way, but biased toward those that I want in higher proportions.

I'm going to see if I can map the urn/marble-painting problem back onto the absent-minded driver problem.