Some decisions, such as predicting the winner of a baseball game, are challenging in part because outcomes are probabilistic. When making such decisions, one view is that humans stochastically and selectively retrieve a small set of relevant memories that provides evidence for competing options. We show that optimal performance at test is impossible when retrieving information in this fashion, no matter how extensive training is, because limited retrieval introduces noise into the decision process that cannot be overcome. One implication is that people should be more accurate in predicting future events when trained on idealized rather than on the actual distributions of items. In other words, we predict the best way to convey information to people is to present it in a distorted, idealized form. Idealization of training distributions is predicted to reduce the harmful noise induced by immutable bottlenecks in people's memory retrieval processes. In contrast, machine learning systems that selectively weight (i.e., retrieve) all training examples at test should not benefit from idealization. These conjectures are strongly supported by several studies and supporting analyses. Unlike machine systems, people's test performance on a target distribution is higher when they are trained on an idealized version of the distribution rather than on the actual target distribution. Optimal machine classifiers modified to selectively and stochastically sample from memory match the pattern of human performance. These results suggest firm limits on human rationality and have broad implications for how to train humans tasked with important classification decisions, such as radiologists, baggage screeners, intelligence analysts, and gamblers.