This paper proposes a lower bound for the probability that at least one out of $$n$$ arbitrary events occurs. The information used consists of the first- and second- degree Bonferroni summations in conjunction with $$p_1$$ and $$p_n$$ , where $$p_1$$ is the probability that exactly one event occurs and $$p_n$$ is the probability that all $$n$$ events occur. We prove that the proposed bound is a Fréchet optimal lower bound, which is a criterion difficult to achieve in general. The two additional non-negative terms used in the proposed bound make it at least as good as the Dawson–Sankoff lower bound, a Fréchet optimal degree two lower bound using the first- and second- degree Bonferroni summations only. A numerical example is presented to illustrate that in some cases, the improvement can be substantial.