Interesting - just perusing those links, it sounds like a multi-armed bandit problem, in which you reason that if something has worked out before, you should tilt your bets more in that direction. In the context of the urn model, you'd return more balls of the same color for every successful draw. In the context of medicine, you can balance between proving or disproving a treatment effect and actually supplying that treatment to the test subjects who need them.
Relatedly, there's a Bayesian interpretation to overweighting successful past draws. A model where you return one extra ball of the same color to the urn gets you a Dirichlet-multinomial distribution, which is a die-roll distribution where the weights to each face are not known for sure, but are given a probability distribution and revised with observed evidence. In other words: here's an n-sided die, I don't know its weightings, but as I observe outcomes I'll update my beliefs that the sides that come up are more favorably weighted. The number of balls in the urn you start with correspond to your priors; only 1 ball of each color means a very weak belief that it's a fair die, 1000 balls of each color means a strong belief, unequal numbers mean that you start off believing it's weighted.