For argumentative purposes (if trying to reach a consensus with someone who claims to be 95% sure that the coin is 'fair' and accepts the '45-55% heads is fair'

For argumentative purposes (if trying to reach a consensus with someone who claims to be 95% sure that the coin is 'fair' and accepts the '45-55% heads is fair' definition), you would concentrate most of your probability mass there, and find a mutually agreeable shape. The 'it's not shaped like a sphere' thing suggests that some people would take into account visual information about the object to inform their priors, and be prone to rejecting the fitness of the uniform. That may conflict with your sensibilities if you expect very strange coins, but if the priors aren't too crazily bad, if you have enough time for a few more coin flips, and the coin isn't particularly brittle, what's an order of magnitude between friends? Edit: And for formal mathy purposes, ask someone else :)