It is an ongoing debate, what properties of visualizations increase people’s performance when solving Bayesian reasoning tasks. In the discussion of the properties of two visualizations, i.e., the tree diagram and the unit square, we emphasize how both visualizations make relevant subset relations transparent. Actually, the unit square with natural frequencies reveals the subset relation that is essential for the Bayes’ rule in a numerical and geometrical way whereas the tree diagram with natural frequencies does it only in a numerical way. Accordingly, in a first experiment with 148 university students, the unit square outperformed the tree diagram when referring to the students’ ability to quantify the subset relation that must be applied in Bayes’ rule. As hypothesized, in a second experiment with 143 students, the unit square was significantly more effective when the students’ performance in tasks based on Bayes’ rule was regarded. Our results could inform the debate referring to Bayesian reasoning since we found that the graphical transparency of nested sets could explain these visualizations’ effect.

Bayes’ formula is a fundamental statistical method for inference judgments in uncertain situations used by both laymen and professionals. However, since people often fail in situations where Bayes’ formula can be applied, how to improve their performance in Bayesian situations is a crucial question. We based our research on a widely accepted beneficial strategy in Bayesian situations, representing the statistical information in the form of natural frequencies. In addition to this numerical format, we used five visualizations: a 2 × 2-table, a unit square, an icon array, a tree diagram, and a double-tree diagram. In an experiment with 688 undergraduate students, we empirically investigated the effectiveness of three graphical properties of visualizations: area-proportionality, use of discrete and countable statistical entities, and graphical transparency of the nested-sets structure. We found no additional beneficial effect of area proportionality. In contrast, the representation of discrete objects seems to be beneficial. Furthermore, our results show a strong facilitating effect of making the nested-sets structure of a Bayesian situation graphically transparent. Our results contribute to answering the questions of how and why a visualization could facilitate judgment and decision making in situations of uncertainty.

Scanning tunneling microscopy, low-energy electron diffraction, and density-functional theory (DFT) calculations have been applied to study misfit-induced structures of the surface and interface of thin ${\mathrm{ZrO}}_2$ films on $\mathrm{Pt}\left(111\right)$. The ${\mathrm{ZrO}}_2$ films have been prepared by reactive deposition of Zr in an ${\mathrm{O}}_2$ atmosphere as well as by postannealing in an ${\mathrm{O}}_2$ atmosphere. Several structures of the surface and the interface have been observed depending on film thickness and preparation. After deposition, the films display a ${\mathrm{ZrO}}_2\left(111\right)\text{−}(1\times 1)$ structure which shows a tendency towards a continuous rotation within an angle range of $\pm 6°$. At submonolayer coverage, a $(4\times 4)$ structure is revealed which fits to a $(5\times 5)$ or a $(3\sqrt{3}\times 3\sqrt{3})$ interface structure of the $\mathrm{Pt}\left(111\right)$ substrate. Annealing ${\mathrm{ZrO}}_2$ films at temperatures around $950\mathrm{K}$ yields domains of a slightly expanded ${\mathrm{ZrO}}_2\left(111\right)\text{−}(2\times 2)$ structure which fits to a $\mathrm{Pt}\left(111\right)\text{−}(\sqrt{7}\times \sqrt{7})$ interface structure. Further annealing leads to the formation of ${\mathrm{ZrO}}_2\left(111\right)$ domains rotated by angles of $\pm 6.6°$. The domains form ${\mathrm{ZrO}}_2\left(111\right)\text{−}(2\sqrt{3}\times 2\sqrt{3})$ structures which are commensurate to a $(\sqrt{19}\times \sqrt{19})R\pm 36.6°$ superstructure of the $\mathrm{Pt}\left(111\right)$ interface. The $(2\sqrt{3}\times 2\sqrt{3})$ domains are favored in comparison to the $(2\times 2)$ domains and grow at the expense of the latter. Upon annealing the ${\mathrm{ZrO}}_2$ films in ultrahigh vacuum, the $(2\times 2)$ domains disappear whereas the $(2\sqrt{3}\times 2\sqrt{3})$ domains are observed even at temperatures of $1300\mathrm{K}$. Additionally at these high temperatures, a noncommensurate ${\mathrm{Zr}}_x{\mathrm{O}}_y{\mathrm{Pt}}_z$ phase is formed due to $\mathrm{Zr}∕\mathrm{Pt}$ interdiffusion. DFT calculations yield information on the energetics, the interlayer distances, and the misfit-induced buckling of the individual layers of different ${\mathrm{ZrO}}_2$ structures in dependence on film thickness. The calculations agree well with the main experimental findings.

People often struggle with Bayesian reasoning. However, previous research showed that people’s performance (and rationality) can be supported by the way the statistical information is represented. First, research showed that using natural frequencies instead of probabilities as the format of statistical information significantly increases people’s performance in Bayesian situations. Second, research also revealed that people’s performance increases through using visualization. We have built our paper on existing research in this field. Our main aim was to analyze people’s strategies in Bayesian situations that are erroneous even though statistical information is represented as natural frequencies and visualizations. In particular, we compared two pairs of visualization with similar numerical information (tree diagram vs. unit square, and double-tree diagram vs. 2 × 2-table) concerning their impact on people’s erroneous strategies in Bayesian situations. For this aim, we conducted an experiment with 540 university students. The students were randomly assigned to four conditions defined by the four different visualizations of statistical information. The students were asked to indicate a fraction in response to four Bayesian situations. We documented the numerator and denominator of the students’ responses representing a basic set and a subset in a Bayesian situation. Our results showed that people’s erroneous strategies are highly dependent on visualization. A central finding was that the visualization’s characteristic of making the nested-sets structure of a Bayesian situation transparent has a facilitating effect on people’s Bayesian reasoning. For example, compared to the unit square, a tree diagram does not explicitly visualize the set-subset relations that are relevant in a Bayesian situation. Accordingly, compared to a unit square, a tree diagram partly hinders people in finding the correct denominator in a Bayesian situation, and, in particular, triggers selecting a wrong numerator. By analyzing people’s erroneous strategies in Bayesian situations, we contribute to investigating approaches to facilitate Bayesian reasoning and to further develop the teaching of Bayesian reasoning.