Challenge Theory (CT), a new approach to decision under risk departs significantly from expected utility, and is based on firmly psychological, rather than economic, assumptions. The paper demonstrates that a purely cognitive-psychological paradigm for decision under risk can yield excellent predictions, comparable to those attained by more complex economic or psychological models that remain attached to conventional economic constructs and assumptions. The study presents a new model for predicting the popularity of choices made in binary risk problems. A CT-based regression model is tested on data gathered from 126 respondents who indicated their preferences with respect to 44 choice problems. Results support CT's central hypothesis, strongly associating between the Challenge Index (CI) attributable to every binary risk problem, and the observed popularity of the bold prospect in that problem (with r=-0.92 and r=-0.93 for gains and for losses, respectively). The novelty of the CT perspective as a new paradigm is illuminated by its simple, single-index (CI) representation of psychological effects proposed by Prospect Theory for describing choice behavior (certainty effect, reflection effect, overweighting small probabilities and loss aversion).