Self-dual set family

A self-dual set family on $[n]$ contains exactly one of $A,A^c$ for every subset $A$. Its relative proportions $a_j$ in complementary uniform levels satisfy $a_{n-j}=1-a_j$. In particular its measure under a uniform random subset is $1/2$.

Suppose a self-dual set family has level proportions $a_j\le j/n$ for $j<n/2$. Subtract the binomial identity $p={\sum }_j\left(\genfrac{}{}{0ex}{}{n}{j}\right)(j/n)p^j(1-p{)}^{n-j}$ from its biased measure of a set family and pair levels $j,n-j$. The difference isEvery term is nonnegative for $p\ge 1/2$. A self-dual intersecting family has the required level bounds by the Erdős-Ko-Rado theorem.

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.