A note on the probability of at least k successes in n correlated binary trials

We obtain the probability distribution of the number of successes in a sequence of correlated binary trials as a function of marginal probabilities and correlation coefficients. It is based on Bahadur's representation of the joint probability distribution of correlated binary trials, truncated to second-order correlations. We provide new results, illustrating the use of this distribution in reliability theory and decision theory.