On non-informativeness in a classical Bayesian inference problem

Not only between frequentists and Bayesians, but also among Bayesians, there is discrepancy on the answer of the essential question: 'Given s successes in n previous trials, what is the probability of success at the next trial (all trials identical and independent)?'. In this paper the Bayesian answer to this problem is discussed considering non-informative prior subjective data. Also some paradoxes related to this problem are discussed. The actual paradox seems to be that our feeling about using subjective information additional to real data that have become available depends on these data, which is not allowed in the Bayesian framework. Key Words: Bernoulli trials, Bayesian statistics, non-informative prior, predictive probability, Hardy's paradox, Bing's paradox, rule of succession.