The Paradigm of Complex Probability and Thomas Bayes’ Theorem

The mathematical probability concept was set forth by Andrey Nikolaevich Kolmogorov in 1933 by laying down a five-axioms system. This scheme can be improved to embody the set of imaginary numbers after adding three new axioms. Accordingly, any stochastic phenomenon can be performed in the set C of complex probabilities which is the summation of the set R of real probabilities and the set M of imaginary probabilities. Our objective now is to encompass complementary imaginary dimensions to the stochastic phenomenon taking place in the “real” laboratory in R and as a consequence to gauge in the sets R, M, and C all the corresponding probabilities. Hence, the probability in the entire set C = R + M is incessantly equal to one independently of all the probabilities of the input stochastic variable distribution in R, and subsequently the output of the random phenomenon in R can be evaluated totally in C. This is due to the fact that the probability in C is calculated after the elimination and subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic phenomenon. We will apply this novel paradigm to the classical Bayes’ theorem in probability theory.

THINKING STRUCTURE OF STUDENTS’ UNDERSTANDING OF PROBABILITY CONCEPT IN TERM OF APOS THEORY

This study aims to analyze the mental structure experienced by students when understanding the concept of probability reviewed from APOS Theory and then suggests a lesson that accommodates the mental structure. APOS theory states that a learner forms a suitable mental structure when interpreting a mathematical concept. This study involved 106 third semester students who enrolled in Probability Theory. The students were given ACE (Activities, Classroom, Exercises) learning cycle treatment. After treatment, students were then given homework assignments that aim to reinforce the learning process. After the sixth week of learning, data were collected through a test. The results of this study are as follows: (1) the mental structure of students towards the concept of opportunity is still at the process level, not at the object-level, (2) Improving the learning of probability concept requires activities to improve verbal understanding, not only in the form of pictures and symbols. The alternative learning treatments are written in this article.

Professional competency: Pre-service mathematics teachers’ understanding toward probability concept

The ability to understand mathematics is a faculty crucial to be possessed by pre-service teachers who will enter into the education sphere. It is one of the professional competencies essential for teachers since satisfactory lessons’ delivery engenders more comprehensible instruction in teachers’ students. Qualitative research employs a descriptive approach relevant to the research purpose, describing mathematics pre-service teachers’ professional competency of understanding the concept toward probability observed based on mathematical abilities; advanced, intermediate, and basic mathematics ability. The subjects are three pre-service teachers having passed discrete mathematics course in a college in Madura. The criteria to select the subjects are the GPA (Grade-point Average) of the last semester and information from the lecturer. It is because unlikely to administer the test due to online learning applied at the college. Findings indicated that the subject with advanced mathematics ability could meet individual concepts, relate concepts and connect concepts with the operations. The subject with intermediate mathematics ability could meet individual concepts, but could not relate to some concepts. However, he could connect concepts with the operations. The subject with basic mathematics ability could not meet individual concepts, relate concepts, and connect concepts with the operations. In terms of the advancement indicator of understanding the concept, the subjects have not attained it since they have not apprehended concept definition well, particularly the probability concept, although the subject with advanced mathematics ability was procedurally prodigious.

On the Probability in General Physics from the Perspective of the Energy Structure

What are the all final possible states that a physical system can reach when some energy is applied to the system? This question can be known as one of the physical questions that relate to the probability in physics. The second law of thermodynamics is known as the base of the probability concept that is raised in modern physics, while this concept is extracted in physical theories by different meanings. As a new approach to investigate the probability, at first, the energy space concept is extracted and then by applying the effects of the second law of thermodynamics on it, the energy structure is presented. The energy structure of the system is a new perspective to investigate the probability, and by using it, the relation between all possible accessible states, when some energy is applied to the system, can be determined.

The Teaching Proposal of Probability in a Brazilian Elementary School Textbook Collection

We present in this article the analysis of the probability teaching proposal ofa textbook collectionused in the initial years of Brazilian elementary school, approved by the National Program of Textbooks (PNLD) in 2016. This work is related to the study group of Didactics of the Mathematics (DDMat). As a theoretical and methodological contribution, we used the anthropological theory of didactic, developed by Yves Chevallard and collaborators, that allows us to model and analyze the choices regarding mathematics and didactics present in the teaching of probability in the collection adopted. By the production and analysis of data, we noticed that some notions regarding probability were proposed from the volume referring to the second year of elementary school, in activities about possibilities. The formalization of the probability concept was presented in the last volume of the collection with activities associated with the idea of ​​fraction, ratio and percentage.

Near-Field Remote Sensing of Surface Velocity and River Discharge Using Radars and the Probability Concept at 10 U.S. Geological Survey Streamgages

Near-field remote sensing of surface velocity and river discharge (discharge) were measured using coherent, continuous wave Doppler and pulsed radars. Traditional streamgaging requires sensors be deployed in the water column; however, near-field remote sensing has the potential to transform streamgaging operations through non-contact methods in the U.S. Geological Survey (USGS) and other agencies around the world. To differentiate from satellite or high-altitude platforms, near-field remote sensing is conducted from fixed platforms such as bridges and cable stays. Radar gages were collocated with 10 USGS streamgages in river reaches of varying hydrologic and hydraulic characteristics, where basin size ranged from 381 to 66,200 square kilometers. Radar-derived mean-channel (mean) velocity and discharge were computed using the probability concept and were compared to conventional instantaneous measurements and time series. To test the efficacy of near-field methods, radars were deployed for extended periods of time to capture a range of hydraulic conditions and environmental factors. During the operational phase, continuous time series of surface velocity, radar-derived discharge, and stage-discharge were recorded, computed, and transmitted contemporaneously and continuously in real time every 5 to 15 min. Minimum and maximum surface velocities ranged from 0.30 to 3.84 m per second (m/s); minimum and maximum radar-derived discharges ranged from 0.17 to 4890 cubic meters per second (m3/s); and minimum and maximum stage-discharge ranged from 0.12 to 4950 m3/s. Comparisons between radar and stage-discharge time series were evaluated using goodness-of-fit statistics, which provided a measure of the utility of the probability concept to compute discharge from a singular surface velocity and cross-sectional area relative to conventional methods. Mean velocity and discharge data indicate that velocity radars are highly correlated with conventional methods and are a viable near-field remote sensing technology that can be operationalized to deliver real-time surface velocity, mean velocity, and discharge.

Development of Probability Concept for Thermal Fatigue Life Cycle Prediction

Furnaces are most commonly used for melting of Ferrous Metals and its alloy materials. Induction furnaces use Electrical Power so that they are more advantageous as no fuel is required. It is a very critical problem to find life span of Induction Melting Furnace Wall under thermal load variation. The life cycle of induction furnace refractory wall is a variable as minor variation is always present due to effect of skill of workers and many other factors. The life cycle of furnace wall will vary minor with some miscellaneous factors and cannot be justified as a single value always. The probability concept is utilized here in the forecast of life cycle calculation to justify the miscellaneous factors effected for the damage of the induction furnace refractory wall. The probability concept initially defines a minimum life of induction furnace wall for a certain case then it is assumed to vary with different probability as given below. So, all the cases of induction furnace wall are having minimum life always but some cases of induction furnace wall are having much longer life. It is due to effect from many miscellaneous factors like skills of workers, efficiency of workers, raw material quality used for construction of wall, tools applied for ramming of it, row material employed for melting, etc.

Arguing and Proving in Vargas´s German-Language Work

Vargas’s work focused on children´s autonomous activities as well as on their intrinsic motivation in mathematics classrooms and was designed in the sense of the so-called genetic method (Ambrus & Vancsó, 2017, p. 7). His goal was to enable learners to discover mathematics in various every-day situations and to find appropriate mathematical models (Ambrus & Vancsó, 2017, p. 10). In a playful way, he wanted to convey methods, models and basic concepts of mathematics and especially the rules of logic. However, a main and basic area of mathematics, which also should be communicated to children from the very beginning, and which is not independent of logics, is the area of arguing and proving in mathematics. The question, how Varga placed and communicated arguments and proofs suitable for 1st-to-4th-grade learners, will be answered by analysing his German-language book Engel, Varga & Walser (1974). In this issue, two of his favourite mathematical areas, namely combinatorics and probability theory are brought to bear. ZDM Subject Classification: E50, K20, K50 Key words and phrases: Reasoning and proving in the mathematics class-room, Combinatorics, Probability concept and probability theory

Measuring Students’ Understanding in Probability Concept between Two Different Cohort using Rasch Measurement

The students’ level of proficiency in any particular course is individually distinctive. Therefore, it is necessary for the educators to be able to address their student’s level of ability in understanding of the course they enrolled. Particularly, educators should be able to design a set of questions which suits the level of their students’ ability. For this reason, this study is concentrated on identifying the level of student’s ability in understanding probability concepts that has been included in the statistics course (STA150: Probability and Statistics 1). This course was enrolled by two groups of students (Group A and Group B) from the Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA Perak branch. Group A enrolled the course in December 2015 until March 2016 whilst Group B enrolled in June until November 2016 sessions. Since the aims of this study are to investigate the difference in students’ conceptual knowledge and understanding of probability concepts, as well as to examine which concepts were found most difficult by the students, hence the Rasch measurement approach was used to explore those aims. An instrument consists of 20 items in a test based on the “Counting Rule” topic was formed by an experienced lecturer to measure the level of student’s ability between the two groups. Based on the findings, it was found that there is a high reliability index of 0.93 (Group A) and 0.88 (Group B) which suggests the suitability of the instrument developed in this study to be replicated to the other samples, even though, the person student’s responses to the items for both groups appeared differently in terms of their difficulties in understanding the probability concepts which represents the student’s ability on this particular topic.