From the self-organization of the cytoskeleton to the synchronous motion of bird flocks, living matter has the extraordinary ability to behave in a concerted manner. The Boltzmann equation for self-propelled particles is frequently used in silico to link a system?s meso- or macroscopic behaviour to the microscopic dynamics of its constituents. But so far such studies have relied on an assumption of simplified binary collisions owing to a lack of experimental data suggesting otherwise. We report here experimentally determined binary-collision statistics by studying a recently introduced molecular system, the high-density actomyosin motility assay. We demonstrate that the alignment induced by binary collisions is too weak to account for the observed ordering transition. The transition density for polar pattern formation decreases quadratically with filament length, indicating that multi-filament collisions drive the observed ordering phenomenon and that a gas-like picture cannot explain the transition of the system to polar order. Our findings demonstrate that the unique properties of biological active-matter systems require a description that goes well beyond that developed in the framework of kinetic theories.