Teaching Probability and Statistics is more than teaching the mathematics. Historically, the math was first developed through analyzing popular games of chance such as involving the rolling of dice. This paper makes the case that the development of the understanding of Probability and Statistics is dependent upon building a “mature” understanding of common random phenomena such as the rolling of dice or the blind drawing of colored balls from an urn. An analysis of verbal protocols of 24 college students, who interact with and describe random phenomena involving the mixture of colored marbles, is presented, using cognitive schema to represent the subjects’ expressed understanding. A cognitive schema representing a “mature” understanding of the random phenomenon is contrasted to a diversity of schema representing observed “immature” understanding. Teaching to explicitly build the mature cognitive schema is proposed.