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Humans frequently make inferences about uncertain future events with limited data. A growing body of work suggests that infants and other primates make surprisingly sophisticated inferences under uncertainty. First, we ask what underlying cognitive mechanisms allow young learners to make such sophisticated inferences under uncertainty. We outline three possibilities, the logic, probabilistic, and heuristics views, and assess the empirical evidence for each. We argue that the weight of the empirical work favors the probabilistic view, in which early reasoning under uncertainty is grounded in inferences about the relationship between samples and populations as opposed to being grounded in simple heuristics. Second, we discuss the apparent contradiction between this early-emerging sensitivity to probabilities with the decades of literature suggesting that adults show limited use of base-rate and sampling principles in their inductive inferences. Third, we ask how these early inductive abilities can be harnessed for improving later mathematics education and inductive inference. We make several suggestions for future empirical work that should go a long way in addressing the many remaining open questions in this growing research area.