= Weighted Bernoulli majority bound
{title2=$\mathbb P(\sum_ic_iZ_i\ge1/2)\ge p$}
Let $c_i\ge0$ sum to one, with no subset sum equal to $1/2$, and let independent <Bernoulli random variables> $Z_i$ all have parameter $p\ge1/2$. The subsets of weight greater than $1/2$ form an intersecting <self-dual set family>. The <complementary-layer bound for biased measure> proves $\mathbb P(\sum_ic_iZ_i\ge1/2)\ge p$. Nonnegative weights ensure that disjoint sets cannot both have weight greater than $1/2$; the no-tie condition supplies self-duality.
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