Calculating probabilities is a crucial task of classical probability theory. Adding supplementary dimensions to nondeterministic experiments will yield a deterministic expression of the theory of probability. This is the novel and original idea at the foundation of my complex probability paradigm. As a matter of fact, probability theory is a stochastic system of axioms in its essence; that means that the phenomena outputs are due to randomness and chance. By adding novel imaginary dimensions to the nondeterministic phenomenon happening in the set R will lead to a deterministic phenomenon and thus a stochastic experiment will have a certain output in the complex probability set and total universe G = C. If the chaotic experiment becomes completely predictable, then we will be fully capable to predict the output of random events that arise in the real world in all stochastic processes. Accordingly, the task that has been achieved here was to extend the random real probabilities set R to the deterministic complex probabilities set and total universe G = C=R+M and this by incorporating the contributions of the set M, which is the complementary imaginary set of probabilities to the set R. Consequently, since this extension reveals to be successful, then an innovative paradigm of stochastic sciences and prognostic was put forward in which all nondeterministic phenomena in R was expressed deterministically in C. This paradigm was initiated and elaborated in my previous 21 publications. Furthermore, this model will be linked to my theory of Metarelativity, which takes into consideration faster-than-light matter and energy. This is what I named “The Metarelativistic Complex Probability Paradigm (MCPP),” which will be developed in the present two chapters 1 and 2.
Calculating probabilities is a crucial task of classical probability theory. Adding supplementary dimensions to nondeterministic experiments will yield a deterministic expression of the theory of probability. This is the novel and original idea at the foundation of my complex probability paradigm. As a matter of fact, probability theory is a stochastic system of axioms in its essence; that means that the phenomena outputs are due to randomness and chance. By adding novel imaginary dimensions to the nondeterministic phenomenon happening in the set R will lead to a deterministic phenomenon and thus a stochastic experiment will have a certain output in the complex probability set and total universe G = C. If the chaotic experiment becomes completely predictable, then we will be fully capable to predict the output of random events that arise in the real world in all stochastic processes. Accordingly, the task that has been achieved here was to extend the random real probabilities set R to the deterministic complex probabilities set and total universe G = C=R+M and this by incorporating the contributions of the set M, which is the complementary imaginary set of probabilities to the set R. Consequently, since this extension reveals to be successful, then an innovative paradigm of stochastic sciences and prognostic was put forward in which all nondeterministic phenomena in R was expressed deterministically in C. This paradigm was initiated and elaborated in my previous 21 publications. Furthermore, this model will be linked to my theory of Metarelativity, which takes into consideration faster-than-light matter and energy. This is what I named “The Metarelativistic Complex Probability Paradigm (MCPP),” which will be developed in the present two chapters 1 and 2.
All our work in classical probability theory is to compute probabilities. The original idea in this research work is to add new dimensions to our random experiment, which will make the work deterministic. In fact, probability theory is a nondeterministic theory by nature; which means that the outcome of the events is due to chance and luck. By adding new dimensions to the event in the real set of probabilities R, we make the work deterministic, and hence a random experiment will have a certain outcome in the complex set of probabilities and total universe G = C. It is of great importance that the stochastic system, like in real-world problems, becomes totally predictable since we will be totally knowledgeable to foretell the outcome of chaotic and random events that occur in nature, for example, in statistical mechanics or in all stochastic processes. Therefore, the work that should be done is to add to the real set of probabilities R the contributions of M, which is the imaginary set of probabilities that will make the event in G = C=R+Mdeterministic. If this is found to be fruitful, then a new theory in statistical sciences and in science, in general, is elaborated and this is to understand absolutely deterministically those phenomena that used to be random phenomena in R. This paradigm was initiated and developed in my previous 21 publications. Moreover, this model will be related to my theory of Metarelativity, which takes into account faster-than-light matter and energy. This is what I called “The Metarelativistic Complex Probability Paradigm (MCPP),” which will be elaborated on in the present two chapters 1 and 2.
All our work in classical probability theory is to compute probabilities. The original idea in this research work is to add new dimensions to our random experiment, which will make the work deterministic. In fact, probability theory is a nondeterministic theory by nature; that means that the outcome of the events is due to chance and luck. By adding new imaginary dimensions to the event in the real set of probabilities R, we make the work deterministic and hence a random experiment will have a certain outcome in the complex set of probabilities and total universe G = C. It is of great importance that the stochastic system, like in the real-world problems, becomes totally predictable since we will be totally knowledgeable to foretell the outcome of chaotic and random events that occur in nature like for example in statistical mechanics or in all stochastic processes. Therefore, the work that should be done is to add to the real set of probabilities R, the contributions of M which is the imaginary set of probabilities which will make the event in G = C = R + M deterministic. If this is found to be fruitful, then a new theory in statistical sciences and in science in general is elaborated and this is to understand absolutely deterministically those phenomena that used to be random phenomena in R. This paradigm was initiated and developed in my previous 25 publications and research works. Moreover, this model will be related to my Theory of Metarelativity which takes into account faster-than-light matter and energy. This is what I called ‘The Metarelativistic Complex Probability Paradigm (MCPP)’ which will be elaborated in the present book.
The crucial job of the theory of classical probability is to compute and assess probabilities. A deterministic expression of probability theory will be achieved by the addition of new dimensions to the stochastic experiments. This is the original and novel idea at the foundation of my paradigm. As a matter of fact, since the events' outcomes are due to randomness and chance, then the theory of probability is a nondeterministic system in its essence. A deterministic experiment and hence a stochastic event will have a certain result in the complex probability set C after encompassing novel imaginary dimensions to the chaotic experiment occurring in the real set R. Thus, we will be fully knowledgeable to predict the outcome of stochastic experiments that arise in the real world in all stochastic processes if the random event becomes completely predictable. Hence, extending the real probabilities set R to the deterministic complex probabilities set C = R + M by including the contributions of the set M which is the imaginary set of probabilities, is the work that has been accomplished here. Therefore, a novel paradigm of stochastic sciences and prognostic was laid down in which all stochastic phenomena in R were expressed deterministically in C since this extension was found to be successful. I coined this original model with the term: “The Complex Probability Paradigm” or CPP for short. Knowing that it was illustrated and initiated in my previous research publications. Henceforth, this original probability paradigm will be applied in this work to Regular Markov Chains and Processes.
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 C 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 chaotic factor, 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. Accordingly, my purpose here is to link my complex probability paradigm to logic. Hence, after adding the time dimension, we will apply this novel paradigm to a newly defined logic that I called ‘Dynamic Logic’.
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 probability axioms of Andrey Nikolaevich Kolmogorov put forward in 1933 can be developed to encompass the set of imaginary numbers after adding to his established five axioms a supplementary three axioms. Therefore, any probabilistic phenomenon can thus be performed in what is now the set of complex probabilities C which is the sum of the real set of probabilities R and the complementary and associated and corresponding imaginary set of probabilities M. The aim here is to compute the complex probabilities by taking into consideration additional novel imaginary dimensions to the phenomenon that occurs in the “real” laboratory. Hence, the corresponding probability in the entire probability set C=R+M is, whatever the random distribution of the input random variable considered in R, permanently and constantly equal to 1. Thus, the result of the stochastic experiment in C can be foretold perfectly and completely. Subsequently, the consequence shows that luck and chance in R is substituted now by absolute determinism in C. Accordingly, this is the consequence of the fact that the probability in C is got by subtracting from the degree of our knowledge of the random system the chaotic factor. Henceforth, I will apply to the established and well-known theory of quantum mechanics my innovative and original Complex Probability Paradigm (CPP) which will yield a completely deterministic expression of quantum theory in the universe of probabilities C=R+M.