Starting with r reds and b blacks, the expected return is
V(r,b) = 2r+b/(r+bCr)
Note that this satisfies
V(r,b) = (r/r+b)*V(r-1,b) + (b/r+b)*V(r,b-1)
Therefore V is a martingale! This means that ANY rational strategy will yield the same expected return. 'Rational' here just means never betting on the rarer colour, and always betting everything when only one colour remains.
Moreover, there exists a risk-free strategy of always betting a proportion (r-b)/(r+b) of current wealth (or likewise (b-r)/(r+b) if there are more blacks remaining), such that one is GUARANTEED a return of $9.08.
