This is interesting psychology. Eventually 10 in a row has to happen. But would you bet 50 cents to win 51? At what point is it Bayesian statistics that tells you not to make the bet? (That somehow the coin is loaded)
If you wish to start with a probability distribution assuming a degree of fairness you can do that and it won't be so prone to fluctuation of the start. I don't know anything online about this offhand but check out "Data Analysis A Bayesian Tutorial" by D. S. Sivia section 2.1.1 for a worked out example.
Once you're doing that, then you need to calculate the optimum amount to bet. If the betting will continue indefinitely then a sensible thing to do would be to bet to maximise the expected logarithm of your bankroll, which you calculate with the Kelley criterion: http://en.wikipedia.org/wiki/Kelly_criterion
If you only get to make one bet then you may just want to maximise expected value.
EDIT: didn't explain how to translate the fairness distribution into win/loss odds because I don't know offhand, you could always simulate.
