This work aims to identify and quantify the biases behind the anomalous behavior of people when they deal with the Three Doors dilemma, which is a really simple but counterintuitive game. Carrying out an artefactual field experiment and proposing eight different treatments to isolate the anomalies, we provide new interesting experimental evidence on the reasons why subjects fail to take the optimal decision. According to the experimental results, we are able to quantify the size and the impact of three main (...) biases that explain the anomalous behavior of participants: Bayesian updating, illusion of control and status quo bias. (shrink)

In teaching statistics in secondary schools and at university, two visualizations are primarily used when situations with two dichotomous characteristics are represented: 2×2 tables and tree diagrams. Both visualizations can be depicted either with probabilities or with frequencies. Visualizations with frequencies have been shown to help students significantly more in Bayesian reasoning problems than probability visualizations do. Because tree diagrams or double-trees (which are largely unknown in school) are node-branch-structures, these two visualizations (compared to the 2×2 table) can even simultaneously (...) display probabilities on branches and frequencies inside the nodes. This is a teaching advantage as it allows the frequency concept to be used to better understand probabilities. However, 2×2 tables and (double-) trees have a decisive disadvantage: While joint probabilities (e.g., P(AB)) are represented in 2×2 tables but no conditional probabilities (e.g., P(A|B)), it is exactly the other way around with (double-) trees. Therefore, a visualization that is equally suitable for the representation of joint probabilities and conditional probabilities is desirable. In this article, we present a new visualization—the frequency net—in which all absolute frequencies and all types of probabilities can be depicted. In addition to a detailed theoretical analysis of the frequency net, we report the results of a study with 249 university students that shows that “net diagrams” can improve reasoning without previous instruction to a similar extent as 2×2 tables and double-trees. Regarding questions about conditional probabilities, frequency visualizations (2×2 table, double-tree, or net diagram with natural frequencies) are consistently superior to probability visualizations. The frequency net performs as well as the frequency double-tree. Only the 2×2 table with frequencies—the only visualization that participants were already familiar with —led to higher performance rates. If, on the other hand, a question about a joint probability had to be answered, all implemented visualizations clearly supported participants’ performance, and no uniform format effect becomes visible. Participants reached the highest performance in the versions with probability 2×2 tables and probability net diagrams. Furthermore, after conducting a detailed error analysis, we report interesting error shifts between the two information formats and the different visualizations and give recommendations for teaching probability. (shrink)

Psychological research on people’s understanding of natural language connectives has traditionally used truth table tasks, in which participants evaluate the truth or falsity of a compound sentence given the truth or falsity of its components in the framework of propositional logic. One perplexing result concerned the indicative conditional if A then C which was often evaluated as true when A and C are true, false when A is true and C is false but irrelevant“ (devoid of value) when A is (...) false (whatever the value of C). This was called the “psychological defective table of the conditional.” Here we show that far from being anomalous the “defective” table pattern reveals a coherent semantics for the basic connectives of natural language in a trivalent framework. This was done by establishing participants’ truth tables for negation, conjunction, disjunction, conditional, and biconditional, when they were presented with statements that could be certainly true, certainly false, or neither. We review systems of three-valued tables from logic, linguistics, foundations of quantum mechanics, philosophical logic, and artificial intelligence, to see whether one of these systems adequately describes people’s interpretations of natural language connectives. We find that de Finetti’s (1936/1995) three-valued system is the best approximation to participants’ truth tables. (shrink)