When the process of cell-fate determination is examined at single-cell resolution, it is often observed that individual cells undergo different fates even when subject to identical conditions. This "noisy" phenotype is usually attributed to the inherent stochasticity of chemical reactions in the cell. Here we demonstrate how the observed single-cell heterogeneity can be explained by a cascade of decisions occurring at the subcellular level. We follow the postinfection decision in bacteriophage lambda at single-virus resolution, and show that a choice between lysis and lysogeny is first made at the level of the individual virus. The decisions by all viruses infecting a single cell are then integrated in a precise (noise-free) way, such that only a unanimous vote by all viruses leads to the establishment of lysogeny. By detecting and integrating over the subcellular "hidden variables," we are able to predict the level of noise measured at the single-cell level.