Hello,
Suppose You have two coins A and B (distributions over head/tail) with different probabilities distributions.
I write A<=B if the probability of head using A is <= than using B.
I want to design a stochastic experiment using A and B (most likely infinite) to test wether A<=B.
The test must be such that there is an event E that have probability 1 just in case A<=B.
Throwing the dices indipendently infinite many times and checking ratios is not a good test in this sense beacuse there is no event E with the desired property.
Do you have any idea?