I think I've sorted out the graphic now. And despite subsequent postings to the forum, I STILL believe that 50% is the correct answer. As the decision tree shows, there are 24 possible outcomes, 12 with probability of 1/18, and 12 of 1/36. They appear in pairs of win and lose, each of the same probability. So the probability of a win must be 1/2.

If someone would show me an error in this tree, I am willing to be convinced that I am wrong.

:D