One of two independent Bernoulli processes (arms) with unknown expectations $\rho$ and $\lambda$ is selected and observed at each of $n$ stages. The selection problem is sequential in that the process which is selected at a particular stage is a function of the results of previous selections as well as of prior information about $\rho$ and $\lambda$. The variables $\rho$ and $\lambda$ are assumed to be independent under the (prior) probability distribution. The objective is to maximize the expected number of successes from the $n$ selections. Sufficient conditions for the optimality of selecting one or the other of the arms are given and illustrated for example distributions. The stay-on-a-winner rule is proved.