The turnover in the stellar initial mass function (IMF) at low mass suggests the presence of two independent mass functions that combine in different ways above and below a characteristic mass given by the thermal Jeans mass in the cloud. In the random sampling model introduced earlier, the Salpeter IMF at intermediate to high mass follows primarily from the hierarchical structure of interstellar clouds, which is sampled by various star formation processes and converted into stars at the local dynamical rate. This power-law part is independent of the details of star formation inside each clump and therefore has a universal character. The flat part of the IMF at low mass is proposed here to result from a second, unrelated, physical process that determines only the probability distribution function for final star mass inside a clump of a given mass, and is independent of both this clump mass and the overall cloud structure. Both processes operate for all potentially unstable clumps in a cloud, regardless of mass, but only the first shows up above the thermal Jeans mass, and only the second shows up below this mass. Analytical and stochastic models of the IMF that are based on the uniform application of these two functions for all masses reproduce the observations well.