Subjective probabilities derived from the perceived randomness of sequences of outcomes

This research continues the longstanding tradition of taking an interdisciplinary approach to studies in probability education. Respondents are presented with sequences of heads and tails derived from flipping a fair coin five times, and asked to consider their chances of occurrence. A new iteration of the comparative likelihood task, which maintains the ratio of heads to tails in all of the sequences presented, provides unique insight into individuals’ perceptions of randomness and associated probabilities. In order to expound the aforementioned insight, this research presents unconventional interpretations of the sample space—organized according to switches, longest run, and switches and longest run, which are all based upon individuals’ verbal descriptions of the sample space—to help situate individuals’ answers and justifications within conventional probability. In doing so, it will be shown that conventionally incorrect responses to the task are not, necessarily, devoid of correct probabilistic thinking. The data for this research is based upon two interrelated questionnaires, denoted Study I and Study II. Answers and justifications from the 56 prospective elementary school mathematics teachers in Study I are employed to develop the second iteration of the questionnaire for Study II, which was given to 239 prospective mathematics teachers (comprised of 163 elementary school teachers and 76 secondary school teachers). To accurately render the data compiled in Study II, an original theoretical framework, entitled the meta-sample-space, will be used with a new method, entitled event-description-alignment, to demonstrate, for the first time, that individuals’ probabilities associated with sequences of outcomes based on perceptions of randomness are in accord with, or model, a subjective-sample-space partitioned according to individuals’ interpretations of the sequence element of the CLT.