Since the only thing that matters (to your expected value) is the overall probability of heads v. tails, the problem is a simple one of parameter estimation. You start with a distribution over p (determined by this thing called the principle of maximum entropy, probably just uniform in this case) and then update it with new evidence (for example, P(p=0.01) drops almost to 0 as soon as you see a head). After 100 heads you get a very spiky function at 1. And yes this is optimal.

For any finite amount of data you won't perfectly break even using a bayesian method, but it's better than all the alternatives, as long as you don't leave out some data.