This study examined students' reasoning about simple repeated choices. Each choice involved ''betting'' on two events, differing in probability. We asked subjects to generate or evaluate alternative strategies such as betting on the most likely event on every trial, betting on it on almost every trial, or employing a ''probability matching'' strategy. Almost half of the college students did not generate or rank strategies according to their expected value, but few subjects preferred a strategy of strict probability matching. High-school students (...) showed greater deviations from expected value than college students. Similar misunderstandings were observed in a choice task involving real (not hypothetical) repeated trials. Large gender differences in prediction strategies and in related computational skills were observed. Subjects who understand the optimal strategy usually do so in terms of independence of successive trials rather than calculation. Some subjects understand the concept of independence but fail to bring it to bear, thinking it can be overridden by intuition or local balancing (representativeness). (shrink)