  In certain finite posets, the expected down-degree of their elements is the
same whether computed with respect to either the uniform distribution or the
distribution weighting an element by the number of maximal chains passing
through it. We show that this coincidence of expectations holds for Cartesian
products of chains, connected minuscule posets, weak Bruhat orders on finite
Coxeter groups, certain lower intervals in Young's lattice, and certain lower
intervals in the weak Bruhat order below dominant permutations. Our tools
involve formulas for counting nearly reduced factorizations in 0-Hecke
algebras; that is, factorizations that are one letter longer than the Coxeter
group length.
