The paper discusses Bayesian convergence when the truth is excluded from the analysis by means of a simple coin-tossing example. In the fair-balance paradox a fair coin is tossed repeatedly. A Bayesian agent, however, holds the a priori view that the coin is either biased towards heads or towards tails. As a result the truth (i.e., the coin is fair) is ignored by the agent. In this scenario the Bayesian approach tends to confirm a false model as the data size goes to infinity. I argue that the fair-balance paradox reveals an unattractive feature of the Bayesian approach to scientific inference and explore a modification of the paradox.