The Framework of the Paradigm of Complex Probability and Monté Carlo Methods, Edition 1

What is it about?

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 non-deterministic 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 C. If the chaotic experiment becomes completely predictable then we will be fully capable of predicting the output of random events that occur 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 C = R + M and this by incorporating the contributions of the set M which is the associated and complementary imaginary set of probabilities to the set R. Hence, the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. Consequently, since this extension was revealed to be successful, then an innovative paradigm of stochastic sciences and prognostic was put forward in which all nondeterministic phenomena in R were expressed deterministically in C. I coined this novel model with the term "The Complex Probability Paradigm (or CPP)" which was initiated and established in my earlier research works. Moreover, this pioneering paradigm will be applied in a creative manner to the stochastic procedures and algorithms of the famous and historical Buffon’s needle method to compute PI, to the renowned neutron shielding problem, and to numerous and various topics that arise in Monté Carlo Methods.

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Why is it important?

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 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 is 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 Monte Carlo Methods 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.

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