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.
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.