Risk and Decision Making: The “Logic” of Probability

(ProQuest: ... denotes formulae omitted.)IntroductionThe reader might think that risk has a well-defined meaning and can be taught straightforwardly. The following discussion illustrates how risk can enrich probability teaching but reminds us that deliberate learning paths have to be designed so that teaching can enhance the ideas of risk in the learners. Modelling requires a different "logic" adapted to the problem being modelled and its context, which seems surprising for many as they do not perceive such diversity in mathematics and are also surprised that probability does not follow logical rules such as transitivity. In an earlier paper on risk in health we found it necessary to define risk right at the beginning of the essay:"By risk we understand a situation with inherent uncertainty about the (future) outcomes, which is related to impact (cost, damage, or benefit). Sometimes expected value is used for comparing several decisions, which are 'at stake'. Risk is used heterogeneously, some refer only to the probability inherent to one adverse outcome without regarding its impact, and others refer only to the adverse outcome. A decision between several choices of action might involve one person, or a decision might be 'shared' between two or more stakeholders, e.g., a patient and a doctor who have to find a decision about the next steps. [...] Risk involves two components and both are prone to subjective interpretations: the judgment of impact is different for different individuals and is even more distinct for a person or an institution with a role different from the patient." (Borovcnik, & Kapadia, 2011a, p.l)However, one reviewer criticized our usage of the term "risk" and stated that by risk only the adverse future outcome can be meant. This would also be reflected by sentences like "The operation bears the risk of "I do not climb in the mountains as the risk of an accident is "No risk - no fun". This shows that there is no unique concept of risk and we became aware that there are other feasible w7ays to define and think about risk instead of a precise definition.Therefore, we start with a discussion of various meanings of risk and also refer to concepts linked to risk and describe situations that involve risk. In the second section we identify statistical notions of risk, which include type I and type II errors of statistical tests and Bayesian risk. The third section is devoted to psychological aspects of risk and perception of risk. We continue with three paradigmatic examples of risk that develop genuine features of risk and enrich the notion of probability by their context. In a further section, elementary approaches to probability are intended to make sense of risk involved in specific situations. A special view on the stakeholders that are possibly brought together in decision making corroborates our view7 that there are many ways to model and describe risk, and they all yield feasible descriptions of the situations involved but may differ in character and in what is regarded as the best decision. Implications for teaching complete the multifaceted character of risk that we project by our considerations.Meanings of RiskIn this section, we deal with various ways to define risk, which is unusual from a mathematical perspective, where definitions are usually agreed. Furthermore, the meaning of risk - regardless of the specific definition - is also influenced by the character of probability information that is used for determining risk. We also go into details about relations between risk and hazard, between risk and uncertainty, and risk and utility. This section concludes with an analysis of situations that involve risk. The constituent parts of such situations (like who takes a risk) influence the interpretation of the input information as well as the meaning of the calculated risk.Attempts to Define RiskApart from the vague use of "risk" in everyday language, risk is perceived and defined also in the scientific context. …