We study a class of preference domains that satisfies the familiar properties of minimal richness, diversity and no-restoration. We show that a specific preference restriction, hybridness, has been embedded in these domains so that the preferences are single-peaked at the "extremes" and unrestricted in the "middle". We also study the structure of strategy-proof and unanimous Random Social Choice Functions on these domains. We show them to be special cases of probabilistic fixed ballot rules (introduced by Ehlers, Peters, and Storcken (2002)).