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.