The ability to make inductive inferences from sparse data is a critical aspect of human learning. However, the properties observed in a sample of evidence depend not only on the true extension of those properties but also on the process by which evidence is sampled. Because neither the property extension nor the sampling process is directly observable, the learner's ability to make accurate generalizations depends on what is known or can be inferred about both variables. In particular, different inferences are licensed if samples are drawn randomly from the whole population (weak sampling) than if they are drawn only from the property's extension (strong sampling). Given a few positive examples of a concept, only strong sampling supports flexible inferences about how far to generalize as a function of the size and composition of the sample. Here we present a Bayesian model of the joint dependence between observed evidence, the sampling process, and the property extension and test the model behaviorally with human infants (mean age: 15 months). Across five experiments, we show that in the absence of behavioral cues to the sampling process, infants make inferences consistent with the use of strong sampling; given explicit cues to weak or strong sampling, they constrain their inferences accordingly. Finally, consistent with quantitative predictions of the model, we provide suggestive evidence that infants' inferences are graded with respect to the strength of the evidence they observe.