INCONSISTENCIES IN STUDENTS REASONING ABOUT PROBABILITY

Subjects were asked to select from among four possible sequences the ' 'most likely'' to result from flipping a coin five times. Contrary to the results of Kahneman and Tversky (1972), the majority of subjects ( 72%) correctly answered that the sequences are equally likely to occur . This result suggests, as does performance on similar NAEP items, tha t most secondary school and college-age students view successive outco mes of a random process as independent. However, in a follow-up questi on, subjects were also asked to select the ''least likely'' result. On ly half the subjects who had answered correctly responded again that t he sequences were equally likely; the others selected one of the seque nces as least likely. This result was replicated in a second study in which 20 subjects were interviewed as they solved the same problems. O ne account of these logically inconsistent responses is that subjects reason about the two questions from different perspectives. When asked to select the most likely outcome, some believe they are being asked to predict what actually will happen, and give the answer ''equally li kely'' to indicate that all of the sequences are possible. This reason ing has been described by Konold (1989) as an ''outcome approach'' to uncertainty. This prediction scheme does not fit questions worded in t erms of the least likely result, and thus some subjects select an inco mpatible answer based on ''representativeness'' (Kahneman & Tversky, 1 972). These results suggest that the percentage of secondary school st udents who understand the concept of independence is much lower than t he latest NAEP results would lead us to believe and, more generally, p oint to the difficulty of assessing conceptual understanding with mult iple-choice items.