{"paper":{"title":"On the Probability a Weighted Bernoulli Sum Exceeds Its Mean","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Aleksa Milojevic, Benny Sudakov","submitted_at":"2026-06-29T13:28:12Z","abstract_excerpt":"Let $w_1, \\dots, w_m$ be positive real weights whose sum is $1$, and let $v_1, \\dots, v_m$ be i.i.d. Bernoulli$(p)$ random variables. If we let $X=\\sum_{i=1}^m w_i v_i$, then we conjecture that for all $0\\leq p\\leq 1/3$ we have \\[\\mathbb{P}\\big[X\\geq \\mathbb{E}[X]\\big]\\geq p.\\] In this short note, we observe a connection of this conjecture with a version of the Manickam-Mikl\\'os-Singhi conjecture, which allows one to prove it for sufficiently small values of $p$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.30287","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.30287/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}