Thanks for the compliment. There is a formula for questions like this, which I explain on my site www.mathproblems.info, see problem 116.
With a bankroll of b units, winning goal of g units, probability of winning p, and probability of losing q your probability of success is ((q/p)b-1)/((q/p)b+g-1). In this case b=20, p=18/37, q=19/37, and g=20, 50, and 100. So for a bankroll of $200 the probability is ((19/18)20-1)/((19/18)40-1) = 0.253252.
For a winning goal of $500 the probability is ((19/18)20-1)/((19/18)70-1) = 0.045293.
For a winning goal of $1,000 the probability is ((19/18)20-1)/((19/18)120-1) = 0.002969.