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Book Publisher International (a part of SCIENCEDOMAIN International)

The Paradigm of Complex Probability and Markov Chains Transition Matrices

Abdo Abou Jaoudé

Published Aug 20, 2024• 0 citations
Publisher

Abstract

In the year 1933, the Russian mathematician Andrey Nikolaevich Kolmogorov put forward the system of axioms of modern probability theory. By adding to Kolmogorov’s original five axioms and an additional three axioms, this established system can be extended to encompass the imaginary set of numbers. Accordingly, the complex probability set C will be created and which is the sum of its corresponding real probability belonging to the real set R and of its corresponding imaginary probability belonging to the imaginary set M. Thus, all random phenomena do not occur now in the real set R but in the general complex set C that encompasses both R and M. Hence, we take into consideration supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory to evaluate the complex probabilities. This is consequently the objective of this novel paradigm. Subsequently, the outcome of the stochastic experiments that follow any probability distribution in R is now predicted perfectly and totally in C and the corresponding probability in the whole set C is always equal to one. Afterward, it follows that luck and chance in R are substituted by absolute determinism in C. Therefore, we evaluate the probability of any probabilistic phenomenon in C by subtracting the chaotic factor from the degree of our knowledge of the random system. My groundbreaking Complex Probability Paradigm (or CPP) will be applied to the well-known theory of Markov Chains Transition Matrices in order to express it perfectly and absolutely in a deterministic way in the universe C = R + M as well as to extend it to the probabilities’ universes M and C.

DOI
10.9734/bpi/mono/978-93-48006-18-9/ch2

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In the year 1933, the Russian mathematician Andrey Nikolaevich Kolmogorov put forward the system of axioms of modern probability theory. By adding to Kolmogorov’s original five axioms and an additional three axioms, this established system can be extended to encompass the imaginary set of numbers. Accordingly, the complex probability set C will be created and which is the sum of its corresponding real probability belonging to the real set R and of its corresponding imaginary probability belonging to the imaginary set M. Thus, all random phenomena do not occur now in the real set R but in the general complex set C that encompasses both R and M. Hence, we take into consideration supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory to evaluate the complex probabilities. This is consequently the objective of this novel paradigm. Subsequently, the outcome of the stochastic experiments that follow any probability distribution in R is now predicted perfectly and totally in C and the corresponding probability in the whole set C is always equal to one. Afterward, it follows that luck and chance in R are substituted by absolute determinism in C. Therefore, we evaluate the probability of any probabilistic phenomenon in C by subtracting the chaotic factor from the degree of our knowledge of the random system. My groundbreaking Complex Probability Paradigm (or CPP) will be applied to the well-known theory of Markov Chains Transition Matrices in order to express it perfectly and absolutely in a deterministic way in the universe C = R + M as well as to extend it to the probabilities’ universes M and C.
The system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set C, which is the sum of the real set R with its corresponding real probability, and the imaginary set M with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex set C instead of the real set R. The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Consequently, the corresponding probability in the whole set C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally in C. Subsequently, it follows that, chance and luck in R is replaced by total determinism in C. Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in C. My innovative Complex Probability Paradigm (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C = R + M as well as to the quantum uncertainty principle in order to verify it and to extend it to the universes M and C.
The system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set C, which is the sum of the real set R with its corresponding real probability, and the imaginary set M with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex set C instead of the real set R. The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Consequently, the corresponding probability in the whole set C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally in C. Subsequently, it follows that, chance and luck in R is replaced by total determinism in C. Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in C. My innovative Complex Probability Paradigm (CPP) will be applied to the established theory of logic in order to express it completely deterministically in the probability universe C = R + M. Therefore, after adding the time dimension, we will relate and join this original paradigm to a newly defined logic that I called ‘Dynamic Logic’ and it will be also implemented to pipeline prognostic with the aim of illustrating CPP and this novel kind of logic.
The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Nikolaevich Kolmogorov. Encompassing new imaginary dimensions with the experiment real dimensions will make the work in the complex probability set totally predictable and with a probability permanently equal to one. This is the original idea in my complex probability paradigm. Therefore, this will make the event in C = R + M absolutely deterministic by adding to the real set of probabilities R the contributions of the imaginary set of probabilities M. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the new prognostic model, we will be able to determine the magnitude of the chaotic factor, the degree of our knowledge, the real and imaginary and complex probabilities in the probability sets R and M and C and which are all subject to chaos and random effects. Hence, we will apply this novel paradigm to Absorbing Markov Chains Theory in order to express it totally and absolutely deterministically in the complex universe C of probabilities.
The set of imaginary numbers is taken into account by extending the probability system of five axioms of Andrey Nikolaevich Kolmogorov which was put forward in 1933. This is achieved by adding three new and supplementary axioms. Hence, any random experiment can thus be performed in the extended complex probability set C = R + M which is the sum of the real set R of real probabilities and the imaginary set M of imaginary probabilities. The objective here is to determine the complex probabilities by encompassing and considering additional new imaginary dimensions to the event that occurs in the “real” laboratory. The outcome of the stochastic phenomenon in C can be foretold perfectly whatever the probability distribution of the input random variable in R is since the corresponding probability in the whole set C is permanently and constantly equal to one. Thus, the consequence that follows indicates that randomness and chance in R is substituted now by absolute determinism in C. This is the result of the fact that the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. This novel complex probability paradigm will be applied to a newly defined logic that I called “Dynamic Logic”.
The concept of mathematical probability was established in 1933 by Andrey Nikolaevich Kolmogorov by defining a system of five axioms. This system can be enhanced to encompass the imaginary numbers set after the addition of three novel axioms. As a result, any random experiment can be executed in the complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. We aim here to incorporate supplementary imaginary dimensions to the random experiment occurring in the “real” laboratory in R and therefore to compute all the probabilities in the sets R, M, and C. Accordingly, the probability in the whole set C = R + M is constantly equivalent to one independently of the distribution of the input random variable in R, and subsequently the output of the stochastic experiment in R can be determined absolutely in C. This is the consequence of the fact that the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. My innovative Complex Probability Paradigm (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C = R + M.
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 calculate in the sets R, M, and C all the corresponding probabilities. Hence, the probability is permanently equal to one in the entire set C = R + M 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 determined perfectly 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. My innovative ‘Complex Probability Paradigm’ (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C = R + M as well as to the quantum entropic uncertainty principle in order to verify it and to extend it to the universes M and C.