Students’ use of definitions of independence across different visualizations

The independence of two events can be defined using either joint probabilities or conditional probabilities. Although mathematically equivalent, these definitions highlight different probabilistic structures that may not be equally transparent and accessible across different visualizations. In this study, 319 secondary school students in Germany solved three probabilistic problems involving two binary events, each presented with a fully completed visualization (2×2 table, double tree, or net diagram) in two different information formats (probabilities or natural frequencies). Students’ performance in correctly analyzing the (in)dependence of events across these versions and their use of the definitions of independence were analyzed. The participants showed a particularly low performance in problems presented with probability double trees, even though the relevant probabilities are explicitly displayed. Moreover, 49% of students consistently applied the joint probability definition of independence across all visualizations. These findings suggest that many students rely on stable rather than flexible (or even adaptive) patterns of definition use, highlighting the need to explicitly connect multiple properties of independence in instruction.