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.


Introduction
The crucial job of the theory of classical probability is to compute and to assess probabilities. A deterministic expression of probability theory can be attained by adding supplementary dimensions to nondeterministic and stochastic experiments. This original and novel idea is at the foundations of my new paradigm of complex probability. In its core, probability theory is a nondeterministic system of axioms that means that the phenomena and experiments outputs are the products of chance and randomness. In fact, a deterministic expression of the stochastic experiment will be realized and achieved by the addition of imaginary new dimensions to the stochastic phenomenon taking place in the real probability set R and hence this will lead to a certain output in the set C of complex probabilities. Accordingly, we will be totally capable to foretell the random events outputs that occur in all probabilistic processes in the real world. This is possible because the chaotic phenomenon becomes completely predictable. Thus, the job that has been successfully completed here was to extend the set of real and random probabilities which is the set R to the complex and deterministic set of probabilities which is C ¼ R þ M. This is achieved by taking into account the contributions of the imaginary and complementary set of probabilities to the set R and that we have called accordingly the set M. This extension proved that it was effective and consequently we were successful to create an original paradigm dealing with prognostic and stochastic sciences in which we were able to express deterministically in C all the nondeterministic processes happening in the 'real' world R. This innovative paradigm was coined by the term "The Complex Probability Paradigm" and was started and established in my seventeen earlier publications and research works [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].
At the end, and to conclude, this research work is organized as follows: After the introduction in section 1, the purpose and the advantages of the present work are presented in section 2. Afterward, in section 3, the extended Kolmogorov's axioms and hence the complex probability paradigm with their original parameters and interpretation will be explained and summarized. Moreover, in section 4, the complex probability paradigm axioms are applied to Bayes' theorem for a discrete binary random variable and for a general discrete uniform random variable and which will be hence extended to the imaginary and complex sets. Additionally, in section 5, the flowchart of the new paradigm will be shown. Furthermore, the simulations of the novel model for a discrete random distribution and for a continuous stochastic distribution are illustrated in section 6. Finally, we conclude the work by doing a comprehensive summary in section 7, and then present the list of references cited in the current research work.

The purpose and the advantages of the current publication
The advantages and the purpose of this current work are to: 1. Extend the theory of classical probability to encompass the complex numbers set, hence to bond the theory of probability to the field of complex variables and analysis in mathematics. This mission was elaborated and initiated in my earlier seventeen papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. 5. Represent and show the graphs of the functions and parameters of the innovative paradigm related to Bayes' theorem. 6. Demonstrate that the classical concept of probability is permanently equal to one in the set of complex probabilities; hence, no randomness, no chaos, no ignorance, no uncertainty, no nondeterminism, no unpredictability, and no disorder exist in: C complex set ð Þ¼R real set ð ÞþM imaginary set À Á : 7. Prepare to implement this creative model to other topics in prognostics and to the field of stochastic processes. These will be the job to be accomplished in my future research publications.
Concerning some applications of the novel founded paradigm and as a future work, it can be applied to any nondeterministic phenomenon using Bayes' theorem whether in the continuous or in the discrete cases. Moreover, compared with existing literature, the major contribution of the current research work is to apply the innovative paradigm of complex probability to Bayes' theorem. The next figure displays the major purposes and goals of the Complex Probability Paradigm (CPP) (Figure 1).

The original Andrey Nikolaevich Kolmogorov system of axioms
The simplicity of Kolmogorov's system of axioms may be surprising. Let E be a collection of elements {E 1 , E 2 , … } called elementary events and let F be a set of subsets of E called random events [18][19][20][21][22]. The five axioms for a finite set E are: Axiom 1: F is a field of sets. Axiom 2: F contains the set E. Axiom 3: A non-negative real number P rob (A), called the probability of A, is assigned to each set A in F. We have always 0 ≤ P rob (A) ≤ 1.
Axiom 4: P rob (E) equals 1. Axiom 5: If A and B have no elements in common, the number assigned to their union is: hence, we say that A and B are disjoint; otherwise, we have: And we say also that: P rob A ∩ B ð Þ¼P rob A ð Þ Â P rob B=A ð Þ¼P rob B ð Þ Â P rob A=B ð Þ which is the conditional probability. If both A and B are independent then: P rob A ∩ B ð Þ¼P rob A ð Þ Â P rob B ð Þ. Moreover, we can generalize and say that for N disjoint (mutually exclusive) events A 1 , A 2 , … , A j , … , A N (for 1 ≤ j ≤ N), we have the following additivity rule: And we say also that for N independent events A 1 , A 2 , … , A j , … , A N (for 1 ≤ j ≤ N), we have the following product rule:

Adding the Imaginary Part M
Now, we can add to this system of axioms an imaginary part such that: Axiom 6: Let P m ¼ i Â 1 À P r ð Þbe the probability of an associated complementary event in M (the imaginary part) to the event A in R (the real part). It follows that P r þ P m =i ¼ 1 where i is the imaginary number with i ¼ ffiffiffiffiffiffi À1 p or i 2 ¼ À1. Axiom 7: We construct the complex number or vector z ¼ P r þ P m ¼ P r þ i 1 À P r ð Þhaving a norm z j j such that: Axiom 8: Let Pc denote the probability of an event in the complex probability universe C where C ¼ R þ M. We say that Pc is the probability of an event A in R with its associated event in M such that: 2iP r P m and is always equal to 1: We can see that by taking into consideration the set of imaginary probabilities we added three new and original axioms and consequently the system of axioms defined by Kolmogorov was hence expanded to encompass the set of imaginary numbers. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]

A concise interpretation of the original paradigm
As a summary of the new paradigm, we declare that in the universe R of real probabilities we have the degree of our certain knowledge is unfortunately incomplete and therefore insufficient and unsatisfactory, hence we encompass in our analysis the set C of complex numbers which integrates the contributions of both the real set R of probabilities and its complementary imaginary probabilities set that we have called accordingly M [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Subsequently, a perfect and an absolute degree of our knowledge is obtained and achieved in the universe of probabilities C ¼ R þ M because we have constantly Pc = 1. In fact, a sure and certain prediction of any random phenomenon is reached in the universe C because in this set, we eliminate and subtract from the measured degree of our knowledge the computed chaotic factor. Consequently, this will lead to in the universe C a probability permanently equal to one as it is shown in the following equation: Pc 2 = DOKÀ Chf = DOK + MChf = 1 = Pc deduced from the complex probability paradigm. Moreover, various discrete and continuous stochastic distributions illustrate in my seventeen previous research works this hypothesis and innovative and original model. The figure that follows shows and summarizes the Extended Kolmogorov Axioms (EKA) or the Complex Probability Paradigm (CPP) (Figure 2).

The probabilities and the conditional probabilities
We define the probabilities for the binary random variable A as follows [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]: A is an event occurring in the real probabilities set R such that: P rob A ð Þ ¼ P r . The corresponding associated imaginary complementary event to the event A in the probabilities set M is the event B such that: P rob B ð Þ ¼ P m ¼ i 1 À P r ð Þ. The real complementary event to the event A in R is the event A such that: The EKA or the CPP diagram.
The imaginary complementary event to the event B in M is the event B such that: Furthermore, we can deduce from CPP the following: Þ, that means if the event B occurs in M then the event A, which is its real correspondent and associated event, occurs in R: ð Þ, that means if the event A occurs in R then the event B, which is its imaginary correspondent and associated event, occurs in M: , that means if the event B occurs in M then the event A, which is its real correspondent and associated event, occurs in R: , that means if the event A occurs in R then the event B, which is its imaginary correspondent and associated event, occurs in M: , that means if the event B occurs in M then the event A, which is its real complementary event, occurs in R: , that means if the event A occurs in R then the event B, which is its imaginary complementary event, occurs in M:

The relations to Bayes' theorem
Another form of Bayes' theorem for two competing statements or hypotheses that is, a binary random variable, is in the probability set R equal to: For an epistemological interpretation: For proposition A and evidence or background B, • P rob A ð Þ is the prior probability, the initial degree of belief in A.
• P rob A À Á is the corresponding initial degree of belief in not-A, that A is false • P rob B=A ð Þis the conditional probability or likelihood, the degree of belief in B given that proposition A is true.
• P rob B=A À Á is the conditional probability or likelihood, the degree of belief in B given that proposition A is false.
• P rob A=B ð Þis the posterior probability, the probability of A after taking into account B.
Therefore, in CPP and hence in C ¼ R þ M, we can deduce the new forms of Bayes' theorem for the case considered as follows: and this independently of the distribution of the binary random variables A in R and correspondingly of B in M: And, its corresponding Bayes' relation in M is: and this independently of the distribution of the binary random variables A in R and correspondingly of B in M. Note that N = 2 corresponds to the binary random variable considered in this case. Similarly, and this independently of the distribution of the binary random variables A in R and correspondingly of B in M: And, its corresponding Bayes' relation in M is: and this independently of the distribution of the binary random variables A in R and correspondingly of B in M: Moreover, and this independently of the distribution of the binary random variables A in R and correspondingly of B in M: And, its corresponding Bayes' relation in M is: and this independently of the distribution of the binary random variables A in R and correspondingly of B in M:

Furthermore,
and this independently of the distribution of the binary random variables A in R and correspondingly of B in M: And, its corresponding Bayes' relation in M is: and this independently of the distribution of the binary random variables A in R and correspondingly of B in M: Since the complex random vector in CPP is z ¼ P r þ P m ¼ P r þ i 1 À P r ð Þthen: Therefore, the resultant complex random vector in CPP is: where N = 2 corresponds to the binary random variable considered in this case. And, To interpret the results obtained, that means that the two probabilities sets R and M are not only associated and complementary and dependent but also equiprobable, which means that there is no preference of considering one probability set on another. Both R and M have the same chance of 0.5 = 1/2 to be chosen in the complex probabilities set C ¼ R þ M. Since That means that the probability in the set C ¼ R þ M is equal to 1, just as predicted by CPP ( Table 1).

The probabilities of dependent and of joint events in
Additionally, we have:

Probability Sets Event Probability Complementary Event Probability
In And, In addition, And, Table 2). 11 The Paradigm of Complex Probability and Thomas Bayes' Theorem DOI: http://dx.doi.org /10.5772/intechopen.98340 Finally, we can directly notice that:

The relations to CPP parameters
The complex random vector where N = 2 corresponds to the binary random variable that we have studied in this case. Thus, The Degree of our knowledge or DOK z1 of z 1 is: The Degree of our knowledge or DOK Z of Z N is: The Chaotic Factor or Chf z1 of z 1 is: The Chaotic Factor or Chf z2 of z 2 is: j. The Magnitude of the Chaotic Factor or MChf z2 of z 2 is: j . The Magnitude of the Chaotic Factor or MChf Z of Z N is: j¼ À0:5 j j¼ 0:5 Table 2.

Sets and Events Sets Intersection Sets Union
The table of the probabilities of dependent and of joint events in C ¼ R þ M: The probability Pc z1 in C ¼ R þ M of z 1 is: The probability Pc z2 in C ¼ R þ M of z 2 is: It is important to note here that all the results of the calculations done above confirm the predictions made by CPP.

Bayes' theorem and CPP and the contingency tables
See Tables 3-7.

The probabilities and the conditional probabilities
Let us consider here a discrete uniform random distribution in the probability set R to illustrate the results obtained for the new Bayes' theorem when related to CPP.
A j is an event occurring in the real probabilities set R such that: The corresponding associated imaginary complementary event to the event A j in the probabilities set M is the event B j such that: The real complementary event to the event A j in R is the event A j such that:  The imaginary complementary event to the event B j in M is the event B j such that: We have also, as derived from CPP that: that means if the event B j occurs in M then the event A j , which is its real complementary event, occurs in R: , that means if the event A j occurs in R then the event B j , which is its imaginary complementary event, occurs in M: that means if the event B j occurs in M then the event A j , which is its real complementary event, occurs in R: that means if the event A j occurs in R then the event B j , which is its imaginary complementary event, occurs in M:

The relations to Bayes' theorem
Bayes' theorem for N competing statements or hypotheses that is, for N random variables, is in the probability set R equal to: Therefore, in CPP and hence in C ¼ R þ M, we can deduce the new forms of Bayes' theorem for the case considered as follows: And, its corresponding Bayes' relation in M is: And, its corresponding Bayes' relation in M is: And, its corresponding Bayes' relation in M is: And, its corresponding Bayes' relation in M is: Therefore, the resultant complex random vectors in CPP of the uniform discrete random distribution are: Analogously, The Monte Carlo Methods -Recent Advances, New Perspectives and Applications That means that the probability in the set C ¼ R þ M is equal to 1, just as predicted by CPP.
Additionally, we have: Moreover, we have:

The relations to CPP parameters
The first complex random vector is: Therefore, the first resultant complex random vector is: The second complex random vector is: Therefore, the second resultant complex random vector is: And, The Degree of our knowledge or DOK z j of z j is: The Degree of our knowledge or DOK z * j of z * j is: The Degree of our knowledge or DOK Z U of Z U N is: The Degree of our knowledge or DOK Z * U of Z * U N is: The Chaotic Factor or Chf z j of z j is: The Chaotic Factor or Chf Z U of Z U N is: The Magnitude of the Chaotic Factor or MChf z j of z j is: The Magnitude of the Chaotic Factor or MChf z * j of z * j is: The Magnitude of the Chaotic Factor or MChf Z U of Z U N is: The Magnitude of the Chaotic Factor or The probability Pc z j in C ¼ R þ M of z j is: It is important to note here that all the results of the calculations done above confirm the predictions made by CPP.

Flowchart of the complex probability and Bayes' theorem prognostic model
The following flowchart summarizes all the procedures of the proposed complex probability prognostic model where X is between the lower bound L b and the upper bound U b :

The new paradigm applied to discrete and continuous stochastic distributions
In this section, the simulation of the novel CPP model for a discrete and a continuous random distribution will be done. Note that all the numerical values found in the paradigm functions analysis for all the simulations were computed using 22 the 64-Bit MATLAB version 2021 software. It is important to mention here that two important and well-known probability distributions were considered although the original CPP model can be applied to any stochastic distribution beside the studied random cases below. This will lead to similar results and conclusions. Hence, the new paradigm is successful with any discrete or continuous random case.

Simulation of the discrete binomial probability distribution
The probability density function (PDF) of this discrete stochastic distribution is: I have taken the domain for the binomial random variable to be: Taking in our simulation N ¼ 10 and p þ q ¼ 1, p ¼ q ¼ 0:5 then: The mean of this binomial discrete random distribution is: The mode for this symmetric distribution is = 5 = Md = μ. The cumulative distribution function (CDF) is: The real probability P rj x ð Þ is: The imaginary complementary probability P mj x ð Þ to P rj x ð Þ is: 10 C k p k q 10Àk , The real complementary probability P * rj x ð Þ to P rj x ð Þ is: The imaginary complementary probability P * mj x ð Þ to P mj x ð Þ is: The complex probability or random vectors are: The Degree of Our Knowledge of z j x ð Þ: The Monte Carlo Methods -Recent Advances, New Perspectives and Applications DOK j x ð Þ is equal to 1 when P rj x ð Þ ¼ P rj L b ¼ 0 ð Þ¼0 and when P rj x ð Þ ¼ P rj U b ¼ 10 ð Þ¼1. The Degree of Our Knowledge of z * j x ð Þ: The Chaotic Factor of z * j x ð Þ: The Magnitude of the Chaotic Factor of z j x ð Þ: . The Magnitude of the Chaotic Factor of z * j x ð Þ: Þ , the probability expressed in the complex probability set C ¼ R þ M is the following: Hence, the prediction of all the probabilities and of Bayes' theorem in the universe C ¼ R þ M is permanently certain and perfectly deterministic (Figure 3).

The Complex Probability Cubes.
In the first cube (Figure 4)  have a minimum at the point K (DOK = 0.5, Chf = À0.5, X = 5). The point L corresponds to (DOK = 1, Chf = 0, X = U b = 10). The three points J, K, L are the same as in Figure 3.
In the second cube (Figure 5), we can notice the simulation of the real probability P r (X) in R and its complementary real probability P m (X)/i in R also in terms of the random variable X for the binomial probability distribution. The thick line in cyan is the projection of the plane Pc 2 (X) = P r (X) + P m (X)/i = 1 = Pc(X) on the plane X = L b = lower bound of X = 0. This thick line starts at the point (P r = 0, P m /i = 1) and ends at the point (P r = 1, P m /i = 0). The red curve represents P r (X) in the plane P r (X) = P m (X)/i in light grey. This curve starts at the point J (P r = 0, P m /i = 1, X = L b = lower bound of X = 0), reaches the point K (P r = 0.5, P m /i = 0.5, X = 5), and gets at the end to L (P r = 1, P m /i = 0, X = U b = upper bound of X = 10). The blue curve represents P m (X)/i in the plane in cyan P r (X) + P m (X)/i = 1 = Pc(X). Notice the importance of the point K which is the intersection of the red and blue curves at X = 5 and when P r (X) = P m (X)/i = 0.5. The three points J, K, L are the same as in Figure 3.
In the third cube (Figure 6), we can notice the simulation of the complex probability z(X) in C ¼ R þ M as a function of the real probability P r (X) = Re(z) in R and of its complementary imaginary probability P m (X) = i Â Im(z) in M, and this in terms of the random variable X for the binomial probability distribution. The red curve represents P r (X) in the plane P m (X) = 0 and the blue curve represents P m (X) in the plane P r (X) = 0. The green curve represents the complex probability z(X) = P r (X) + P m (X) = Re(z) + i Â Im(z) in the plane P r (X) = iP m (X) + 1 or z(X) plane in cyan. The curve of z(X) starts at the point J (P r = 0, P m = i, X = L b = lower bound of X = 0) and ends at the point L (P r = 1, P m = 0, X = U b = upper bound of X = 10). The thick line in cyan is P r (X = L b = 0) = iP m (X = L b = 0) + 1 and it is the projection of the z(X) curve on the complex probability plane whose equation is X = L b = 0. This projected thick line starts at the point J (P r = 0, P m = i, X = L b = 0) and ends at the point (P r = 1, P m = 0, X = L b = 0). Notice the importance of the point K corresponding to X = 5 and z = 0.5 + 0.5i when P r = 0.5 and P m = 0.5i. The three points J, K, L are the same as in Figure 3.

Simulation of the continuous standard Gaussian normal probability distribution
The probability density function (PDF) of this continuous stochastic distribution is: , for À ∞ < x < ∞ and the cumulative distribution function (CDF) is: The domain for this standard Gaussian normal variable is considered in the simulations to be equal to: x ∈ L b ¼ À4, U b ¼ 4 ½ and I have taken dx ¼ 0:01. In the simulations, the mean of this standard normal random distribution is μ ¼ 0.
The mode for this symmetric distribution is = 0 = Md = μ. The real probability P rj x ð Þ is: The graphs of P r and of P m and of z in terms of X for this binomial probability distribution.
The imaginary complementary probability P mj x ð Þ to P rj x ð Þ is: The real complementary probability P * rj x ð Þ to P rj x ð Þ is: The imaginary complementary probability P * mj x ð Þ to P mj x ð Þ is: DOK(X) (red) and Chf(X) (green, blue, pink) in different simulation planes. Notice that they all have a minimum at the point K (DOK = 0.5, Chf = À0.5, X = 0). The point L corresponds to (DOK = 1, Chf = 0, X = U b = 4). The three points J, K, L are the same as in Figure 7.
In the second cube (Figure 9), we can notice the simulation of the real probability P r (X) in R and its complementary real probability P m (X)/i in R also in terms of the random variable X for the standard Gaussian normal probability distribution. The thick line in cyan is the projection of the plane Pc 2 (X) = P r (X) + P m (X)/i = 1 = Pc(X) on the plane X = L b = lower bound of X = À4. This thick line starts at the point (P r = 0, P m /i = 1) and ends at the point (P r = 1, P m /i = 0). The red curve represents P r (X) in the plane P r (X) = P m (X)/i in light grey. This curve starts at the point J (P r = 0, P m /i = 1, X = L b = lower bound of X = À4), reaches the point K (P r = 0.5, P m /i = 0.5, X = 0), and gets at the end to L (P r = 1, P m /i = 0, X = U b = upper bound of X = 4). The blue curve represents P m (X)/i in the plane in cyan P r (X) + P m (X)/i = 1 = Pc(X). Notice the importance of the point K which is the intersection of the red and blue curves at X = 0 and when P r (X) = P m (X)/i = 0.5. The three points J, K, L are the same as in Figure 7.
In the third cube (Figure 10), we can notice the simulation of the complex probability z(X) in C ¼ R þ M as a function of the real probability P r (X) = Re(z) in R and of its complementary imaginary probability P m (X) = i Â Im(z) in M, and this in terms of the random variable X for the standard Gaussian normal probability distribution. The red curve represents P r (X) in the plane P m (X) = 0 and the blue curve represents P m (X) in the plane P r (X) = 0. The green curve represents the complex probability z(X) = P r (X) + P m (X) = Re(z) + i Â Im(z) in the plane P r (X) = iP m (X) + 1 or z(X) plane in cyan. The curve of z(X) starts at the point J (P r = 0, P m = i, X = L b = lower bound of X = À4) and ends at the point L (P r = 1, P m = 0, X = U b = upper bound of X = 4). The thick line in cyan is P r (X = L b = À4) = iP m (X = L b = À4) + 1 and it is the projection of the z(X) curve on the complex probability plane whose equation is X = L b = À4. This projected thick line starts at the point J (P r = 0, P m = i, X = L b = À4) and ends at the point (P r = 1, P m = 0, X = L b = À4). Notice the importance of the point K corresponding to X = 0 and z = 0.5 + 0.5i when P r = 0.5 and P m = 0.5i. The three points J, K, L are the same as in Figure 7.

Conclusion and perspectives
In the current research work, the original extended model of eight axioms (EKA) of A. N. Kolmogorov was connected and applied to the classical Bayes' theorem. Thus, a tight link between this theorem and the novel paradigm was achieved. Consequently, the model of "Complex Probability" was more developed beyond the scope of my seventeen previous research works on this topic. Additionally, as it was proved and verified in the novel model, before the beginning of the random phenomenon simulation and at its end we have the chaotic factor (Chf and MChf) is zero and the degree of our knowledge (DOK) is one since the stochastic fluctuations and effects have either not started yet or they have terminated and finished their task on the probabilistic phenomenon. During the execution of the nondeterministic phenomenon and experiment we also have: 0.5 ≤ DOK < 1, À0.5 ≤ Chf < 0, and 0 < MChf ≤ 0.5. We can see that during this entire process we have incessantly and continually Pc 2 = DOK -Chf = DOK + MChf = 1 = Pc, that means that the simulation which behaved randomly and stochastically in the set R is now certain and deterministic in the probability set C ¼ R þ M, and this after adding to the random experiment executed in R the contributions of the set M and hence after eliminating and subtracting the chaotic factor from the degree of our knowledge. Furthermore, the real, imaginary, complex, and deterministic probabilities that correspond to each value of the random variable X have been determined in the three probabilities sets which are R, M, and C by P r , P m , z and Pc respectively. Consequently, at each value of X, the novel Bayes' theorem and CPP parameters P r , P m , P m =i, DOK, Chf, MChf, Pc, and z are surely and perfectly predicted in the complex probabilities set C with Pc maintained equal to one permanently and repeatedly. In addition, referring to all these obtained graphs and executed simulations throughout the whole research work, we are able to quantify and to visualize both the system chaos and stochastic effects and influences (expressed and materialized by Chf and MChf) and the certain knowledge (expressed and materialized by DOK and Pc) of the new paradigm. This is without any doubt very fruitful, wonderful, and fascinating and proves and reveals once again the advantages of extending A. N. Kolmogorov's five axioms of probability and hence the novelty and benefits of this inventive and original model in the fields of prognostics and applied mathematics that can be called truly: "The Complex Probability Paradigm". Furthermore, it is very crucial to state that using CPP, conditional probabilities, and Bayes' theorem, we have linked and joined and bonded the events probabilities sets R with R, M with M, R with M, M with R, R with C, M with C, and C with C using precise and exact mathematical relations and equations. Moreover, it is important to mention here that the novel CPP paradigm can be implemented to any probability distribution that exists in literature as it was shown in the simulation section. This will lead without any doubt to analogous and similar conclusions and results and will confirm certainly the success of my innovative and original model.
As a future and prospective research and challenges, we aim to more develop the novel prognostic paradigm conceived and to implement it to a large set of random and nondeterministic events like for other probabilistic phenomena as in stochastic processes and in the classical theory of probability. Additionally, we will apply CPP Figure 10. The graphs of P r and of P m and of z in terms of X for the standard Gaussian normal probability distribution.