The definition of a "fair coin" is exactly 1/2 probability of heads and of tails. No "opinion" is involved. I am allowing you to assume all coin tosses are truly independent. Clearly, an unfair coin, no matter how tiny its unfairness, could be detected with confidence 1-K, for given K as small as one would like, after a large enough number of tosses. A fair coin, however, could never be proven fair, and then the testing procedure will simply keep going (or, with probability<=K, terminate and wrongly report coin is unfair -- that is what "confidence" means). Gene Salamin's remark about using another (H,T) is not correct, or at least highly inefficient, because previous experiment was subset of new experiment, and his formulas ignore that. So the question is a simple one: how should we best test the fairness of a coin? (With some specified confidence?) But it is not so easy to answer. In fact so far no math-funner besides me even seems to have comprehended the question... But I believe there probably is a unique answer, i.e. a uniquely optimal statistical test, and have sketched how to characterize that answer as a certain constrained variational problem. This characterization is unsatisfying as matters presently stand.