We investigate monotonicity properties of the success probabilities and the total reward when the number of previously observed successes and failures change. Using a well-known Bayesian approach and dynamic programming we give conditions in terms of the covariances of the posterior distributions and in terms of the support of the prior distribution. Special order relations for the number of successes and failures allow a simple and unified treatment of different cases. The results extend some of the investigations of Hengartner/Kalin/Theodorescu [1].