Much inquiry in psycholinguistics has focused on evidence from the N400 and P600 components of the event-related potential (ERP) signal—and a central theoretical challenge in this area is accounting for the so-called “semantic P600”, which involves unexpected patterns in these components relative to traditional theories of the underlying mechanisms. In this paper we present a computational model of the language processing mechanisms underlying these ERP components, which builds on existing psycholinguistic theories in positing a heuristic interpretation stage of processing, but which deviates from existing theories in formulating this heuristic interpretation process as probabilistic selection via a noisy channel model, and in quantifying and accounting for fine-grained variation in statistical and representational properties of individual stimuli. Our model successfully simulates N400 and P600 patterns from eight psycholinguistic experiments, reflecting the full range of N400-only, P600-only, and biphasic N400-P600 effects, and its behaviors shed light on a number of key patterns that have presented challenges for existing theories. The model’s success indicates that a strong account for the processing mechanisms underlying these effects is one in which language comprehension involves a probabilistic heuristic interpretation stage resembling a noisy channel process, feeding into subsequent processes that assess target word fit and reconcile between heuristic and literal interpretations. The model’s success also indicates that these mechanisms are critically sensitive to statistical variation in individual stimuli, and that modeling the effects of this variation is essential to account for the full range of observed effects in language processing.