Exact bounds on the probability of at least k successes in n exchangeable Bernoulli trials as a function of correlation coefficients

We compute the minimum and maximum of the probability of k or more successes in n exchangeable Bernoulli trials as a function of the correlation coefficients. This probability finds wide application in reliability and decision theory. Since the probability is linear in the coefficients, finding the minimum and maximum requires solving linear programming problems. We show that the maximum can be lower than certainty (no certain success), whereas the minimum can be higher than zero (positive residual risk).