Prospective Mathematics Teachers Understanding of Classical and Frequentist Probability

Strengthening the teaching of probability requires an adequate training of prospective teachers, which should be based on the prior assessment of their knowledge. Consequently, the aim of this study was to analyse how 139 prospective Spanish mathematics teachers relate the classical and frequentist approaches to probability. To achieve this goal, content analysis was used to categorize the prospective teachers’ answers to a questionnaire with open-ended tasks in which they had to estimate and justify the composition of an urn, basing their answers on the results of 1000 extractions from the urn. Most of the sample proposed an urn model consistent with the data provided; however, the percentage that adequately justified the construction was lower. Although the majority of the sample correctly calculated the probability of an event in a new extraction and chose the urn giving the highest probability, a large proportion of the sample forgot the previously constructed urn model, using only the frequency data. Difficulties, such as equiprobability bias or not perceiving independence of trials in replacement sampling, were also observed for a small part of the sample. These results should be considered in the organisation of probabilistic training for prospective teachers.

Frequentist Probability

Frequentist probability is historically presented as an attempt both to overcome the limitations of classical conception and to take into account the impressive development of experimental sciences and statistics. It is precisely because of this close link with the statistical sciences that it finds a significant place in teaching. It is also an area in which the use of IT tools is crucial. This approach also makes it possible to calculate the probability of events and from it we derive the same rules examined in the classical definition, since it is sufficient to replace the ratio between the number of favorable cases and the number of possible cases with the limit of the ratio when the repeated tests tend to infinity. One of the fundamental concepts appears in the chapter, that of a random variable capable of describing events and their distribution.

Frequentist probability in Japanese school curricula

Many Japanese school mathematics teachers, policy-makers and researchers believe that probabilistic contents are difficult for most students to understand. In this study, I identify several reasons for the difficulty through an ecological analysis that is a part of a didactic analysis. This task is achieved through three research techniques: (a) constructing a reference epistemological model of probabilistic activities in terms of praxeology, (b) analysing probabilistic contents of Japanese school mathematics textbooks from the reference model and (c) identifying institutional conditions and constraints on the contents using the scale of levels of didactic codetermination. In current Japanese school mathematics, frequentist probability is hardly mentioned, whereas Laplacian probability comprises a large part of the curriculum of probability, although some generic conditions make the frequentist probability viable. This fact is related to the following three constraints: determinationism, theoricism and demathematisation of randomiser.

Defining Probability in Sex Offender Risk Assessment

There is ongoing debate and confusion over using actuarial scales to predict individuals’ risk of sexual recidivism. Much of the debate comes from not distinguishing Frequentist from Bayesian definitions of probability. Much of the confusion comes from applying Frequentist probability to individuals’ risk. By definition, only Bayesian probability can be applied to the single case. The Bayesian concept of probability resolves most of the confusion and much of the debate in sex offender risk assessment. Although Bayesian probability is well accepted in risk assessment generally, it has not been widely used to assess the risk of sex offenders. I review the two concepts of probability and show how the Bayesian view alone provides a coherent scheme to conceptualize individuals’ risk of sexual recidivism.

On the Behrens-Fisher Problem: A Review

The Behrens-Fisher problem arises when one seeks to make inferences about the means of two normal populations without assuming the variances are equal. This paper presents a review of fundamental concepts and applications used to address the Behrens-Fisher problem under fiducial, Bayesian, and frequentist approaches. Methods of approximations to the Behrens-Fisher distribution and a simple Bayesian framework for hypothesis testing are also discussed. Finally, a discussion is provided for the use of generalized p values in significance testing of hypotheses in the presence of nuisance parameters. It is shown that the generalized p values based on a frequentist probability for the Behrens-Fisher problem are numerically the same as those from the fiducial and Bayesian solutions. A table for tests of significance is also included.