Weighted Bernoulli majority bound ($\mathbb{P}({\sum }_i{c}_i{Z}_i\ge 1/2)\ge p$)

Let $c_i\ge 0$ sum to one, with no subset sum equal to $1/2$, and let independent Bernoulli random variables $Z_i$ all have parameter $p\ge 1/2$. The subsets of weight greater than $1/2$ form an intersecting self-dual set family. The complementary-layer bound for biased measure proves $\mathbb{P}({\sum }_i{c}_i{Z}_i\ge 1/2)\ge p$. Nonnegative weights ensure that disjoint sets cannot both have weight greater than $1/2$; the no-tie condition supplies self-duality.