Entropy-driven formation of high entropy alloy (HEA) nanoparticles from metal precursors requires high temperatures and controlled cooling rates. However, several proposed HEA nanoparticle synthesis strategies avoid the high-temperature regime. In our work, we address the question of how single-phase HEA nanoparticles can form at low temperatures. Investigating a system of five noble metal single source precursors, we combine in situ X-ray powder diffraction with multi-edge X-ray absorption spectroscopy to demonstrate that the formation of single-phase nanoparticles is governed by stochastic principles and the inhibition of precursor mobility during the formation process. The proposed formation principle is supported by simulations of the nanoparticle formation in a randomized process, rationalizing the experimentally found differences between two-element and multi-element metal precursor mixtures.