How Many Rounds Should You Expect in Urn Solitaire?

A certain sampling process, concerning an urn with balls of two colors, proposed in 1965 by B.E. Oakley and R.L. Perry, and discussed by Peter Winkler and Martin Gardner, that has an extremely simple answer for the probability, namely the constant function 1/2, has a far more complicated expected duration, that we discover and sketch the proof of. So unlike, for example, the classical gambler's ruin problem, for which both `probability of winning' and `expected duration' have very simple expressions, in this case the expected number of rounds is extremely complicated, and beyond the scope of humans.