There is a growing body of evidence indicating that people often overestimate the similarity between characteristics of random samples and those of the populations from which they are drawn. This paper: reviews studies that have attempted to determine whether the basic heuristic employed in thinking about random samples is passive and descriptive or whether it is deducible from a belief in active balancing; discusses the importance of sample size on judgments about the characteristics of random samples; and examines implications for instruction. For example, work done on sensitivity to sample size suggests that basic concepts and principles must be illustrated with a variety of examples if students are to be able to generalize them appropriately. Also, many statistics textbooks that discuss the Law of Large Numbers attempt to dispel students' belief in the gambler's fallacy; however, they assume that the basic misconception students have is active balancing, and they oppose this mechanism with the notion of "swamping." Current research suggests that such an approach is likely to be unfruitful because the problem is not that students think in terms of an incorrect process mechanism but that they do not think of random sampling in terms of any process model. (Author/JN)