{"paper":{"title":"Lower Bounds on the Probability of a Finite Union of Events","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.PR","authors_text":"Fady Alajaji, Glen Takahara, Jun Yang","submitted_at":"2014-01-22T02:47:08Z","abstract_excerpt":"In this paper, lower bounds on the probability of a finite union of events are considered, i.e. $P\\left(\\bigcup_{i=1}^N A_i\\right)$, in terms of the individual event probabilities $\\{P(A_i), i=1,\\ldots,N\\}$ and the sums of the pairwise event probabilities, i.e., $\\{\\sum_{j:j\\neq i} P(A_i\\cap A_j), i=1,\\ldots,N\\}$. The contribution of this paper includes the following: (i) in the class of all lower bounds that are established in terms of only the $P(A_i)$'s and $\\sum_{j:j\\neq i} P(A_i\\cap A_j)$'s, the optimal lower bound is given numerically by solving a linear programming (LP) problem with $N^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.5543","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}