Open access peer-reviewed chapter

# The Paradigm of Complex Probability and Thomas Bayes’ Theorem

Written By

Abdo Abou Jaoudé

Submitted: 22 April 2021 Reviewed: 11 May 2021 Published: 14 July 2021

DOI: 10.5772/intechopen.98340

From the Edited Volume

## The Monte Carlo Methods - Recent Advances, New Perspectives and Applications

Edited by Abdo Abou Jaoudé

Chapter metrics overview

View Full Metrics

## Abstract

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.

### Keywords

• Chaotic factor
• degree of our knowledge
• complex random vector
• imaginary probability
• probability norm
• complex probability set

“Simple solutions seldom are. It takes a very unusual mind to undertake analysis of the obvious.”

“Nothing in nature is by chance… Something appears to be chance only because of our lack of knowledge.”

Baruch Spinoza.

“Fundamental progress has to do with the reinterpretation of basic ideas.”

“Mathematics, rightly viewed, possesses not only truth but supreme beauty… ”

Bertrand Russell.

## 1. 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=R+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.

## 2. 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].

2. Apply the novel probability axioms and paradigm to the classical Bayes’ theorem.

3. Show that all nondeterministic phenomena can be expressed deterministically in the complex probabilities set which isC.

4. Compute and quantify both the degree of our knowledge and the chaotic factor of all the probabilities in the sets R, M, and C.

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:

Ccomplexset=Rrealset+Mimaginaryset.

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).

## 3. The complex probability paradigm

### 3.1 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 {E1, E2, …} 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 Prob(A), called the probability of A, is assigned to each set A in F. We have always 0 ≤ Prob(A) ≤ 1.

Axiom 4: Prob(E) equals 1.

Axiom 5: If A and B have no elements in common, the number assigned to their union is:

ProbAB=ProbA+ProbB

hence, we say that A and B are disjoint; otherwise, we have:

ProbAB=ProbA+ProbBProbAB

And we say also that: ProbAB=ProbA×ProbB/A=ProbB×ProbA/B which is the conditional probability. If both A and B are independent then: ProbAB=ProbA×ProbB.

Moreover, we can generalize and say that for N disjoint (mutually exclusive) events A1,A2,,Aj,,AN (for 1jN), we have the following additivity rule:

Probj=1NAj=j=1NProbAj

And we say also that for N independent events A1,A2,,Aj,,AN (for 1jN), we have the following product rule:

Probj=1NAj=j=1NProbAj

### 3.2 Adding the Imaginary Part M

Now, we can add to this system of axioms an imaginary part such that:

Axiom 6: Let Pm=i×1Pr be the probability of an associated complementary event in M (the imaginary part) to the event A in R (the real part). It follows that Pr+Pm/i=1 where i is the imaginary number with i=1 or i2=1.

Axiom 7: We construct the complex number or vector z=Pr+Pm=Pr+i1Pr having a norm z such that:

z2=Pr2+Pm/i2.

Axiom 8: Let Pc denote the probability of an event in the complex probability universe C where C=R+M. We say that Pc is the probability of an event A in R with its associated event in M such that:

Pc2=Pr+Pm/i2=z22iPrPmand is always equal to1.

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].

#### 3.2.1 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=R+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: Pc2 = DOK− 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).

## 4. The complex probability paradigm applied to Bayes’ Theorem

### 4.1 The case of a discrete binary random variable

#### 4.1.1 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: ProbA=Pr.

The corresponding associated imaginary complementary event to the event A in the probabilities set M is the event B such that: ProbB=Pm=i1Pr.

The real complementary event to the event A in R is the event A¯ such that:

AA¯=R and AA¯= (mutually exclusive events)

ProbA¯=1ProbA=1Pr=Pm/i=ProbB/i
ProbB=iProbA¯
ProbR=ProbAA¯=ProbA+ProbA¯=Pr+1Pr=1

The imaginary complementary event to the event B in M is the event B¯ such that:

BB¯=M and BB¯= (mutually exclusive events)

ProbB¯=iProbB=iPm=ii1Pr=ii+iPr=iPr=iProbA

ProbA=ProbB¯/i=iProbB¯ since 1/i=i.

ProbM=ProbBB¯=ProbB+ProbB¯=Pm+iPm=i

ProbR=ProbM/i = 1, just as predicted by CPP.

We have also, as derived from CPP that:

ProbA/B=ProbA=Pr, that means if the event B occurs in M then the event A, which is its real complementary event, occurs in R.

ProbB/A=ProbB=Pm, that means if the event A occurs in R then the event B, which is its imaginary complementary event, occurs in M.

Furthermore, we can deduce from CPP the following:

ProbA/B¯=iPr/i=Pr=ProbA, that means if the event B¯ occurs in M then the event A, which is its real correspondent and associated event, occurs in R.

ProbB/A¯=i1Pr=Pm=ProbB, that means if the event A¯ occurs in R then the event B, which is its imaginary correspondent and associated event, occurs in M.

ProbA¯/B=i1Pr/i=1Pr=ProbA¯, that means if the event B occurs in M then the event A¯, which is its real correspondent and associated event, occurs in R.

ProbB¯/A=iPr=iPm=ProbB¯, that means if the event A occurs in R then the event B¯, which is its imaginary correspondent and associated event, occurs in M.

ProbA¯/B¯=1iPr/i=1Pr=ProbA¯, that means if the event B¯ occurs in M then the event A¯, which is its real complementary event, occurs in R.

ProbB¯/A¯=ii1Pr=iPr=ProbB¯, that means if the event A¯ occurs in R then the event B¯, which is its imaginary complementary event, occurs in M.

#### 4.1.2 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:

ProbA/B=ProbB/AProbAProbB=ProbB/AProbAProbB/AProbA+ProbB/A¯ProbA¯

For an epistemological interpretation:

For proposition A and evidence or background B,

• ProbA is the prior probability, the initial degree of belief in A.

• ProbA¯ is the corresponding initial degree of belief in notA, that A is false

• ProbB/A is the conditional probability or likelihood, the degree of belief in B given that proposition A is true.

• ProbB/A¯ is the conditional probability or likelihood, the degree of belief in B given that proposition A is false.

• ProbA/B is the posterior probability, the probability of A after taking into account B.

Therefore, in CPP and hence in C=R+M, we can deduce the new forms of Bayes’ theorem for the case considered as follows:

ProbA/B=ProbB/AProbAProbB=ProbBProbAProbB=PmPrPm=Pr=ProbA=ProbB/AProbAProbB/AProbA+ProbB/A¯ProbA¯=ProbBProbAProbBProbA+ProbBProbA¯=PmPrPmPr+Pm1Pr=PmPrPmPr+PmPmPr=PmPrPm=Pr=ProbA

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:

ProbB/A=ProbA/BProbBProbA=ProbAProbBProbA=PrPmPr=Pm=ProbB=iN1ProbA/BProbBProbA/BProbB+ProbA/B¯ProbB¯=i21ProbAProbBProbAProbB+ProbAProbB¯=iPrPmPrPm+PriPm=iPrPmPrPm+iPrPrPm=iPrPmiPr=iPmi=Pm=ProbB

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,

ProbA¯/B¯=ProbB¯/A¯ProbA¯ProbB¯=ProbB¯ProbA¯ProbB¯=iPr1PriPr=1Pr=ProbA¯=N1ProbB¯/A¯ProbA¯ProbB¯/A¯ProbA¯+ProbB¯/AProbA=21ProbB¯ProbA¯ProbB¯ProbA¯+ProbB¯ProbA=iPr1PriPr1Pr+iPrPr=iPr1PriPriPr2+iPr2=iPr1PriPr=1Pr=ProbA¯

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:

ProbB¯/A¯=ProbA¯/B¯ProbB¯ProbA¯=ProbA¯ProbB¯ProbA¯=1PriPr1Pr=iPr=iPm=PB¯=iProbA¯/B¯ProbB¯ProbA¯/B¯ProbB¯+ProbA¯/BProbB=iProbA¯ProbB¯ProbA¯ProbB¯+ProbA¯ProbB=i1PriPr1PriPr+1Pri1Pr=iiPriPr+i1Pr=iiPriPr+iiPr=iiPri=iPr=iPm=PB¯

and this independently of the distribution of the binary random variables A in R and correspondingly of B in M.

Moreover,

ProbA/B¯=ProbB¯/AProbAProbB¯=ProbB¯ProbAProbB¯=iPrPriPr=Pr=ProbA=ProbB¯/AProbAProbB¯/AProbA+ProbB¯/A¯ProbA¯=ProbB¯ProbAProbB¯ProbA+ProbB¯ProbA¯=iPrPriPrPr+iPr1Pr=iPrPriPr2+iPriPr2=iPrPriPr=Pr=ProbA

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:

ProbB/A¯=ProbA¯/BProbBProbA¯=ProbA¯ProbBProbA¯=1Pri1Pr1Pr=i1Pr=ProbB=iN1ProbA¯/BProbBProbA¯/BProbB+ProbA¯/B¯ProbB¯=i21ProbA¯ProbBProbA¯ProbB+ProbA¯ProbB¯=i1Pri1Pr1Pri1Pr+1PriPr=ii1Pri1Pr+iPr=ii1PriiPr+iPr=ii1Pri=i1Pr=ProbB

and this independently of the distribution of the binary random variables A in R and correspondingly of B in M.

Furthermore,

ProbA¯/B=ProbB/A¯ProbA¯ProbB=ProbBProbA¯ProbB=Pm1PrPm=1Pr=ProbA¯=N1ProbB/A¯ProbA¯ProbB/A¯ProbA¯+ProbB/AProbA=21ProbBProbA¯ProbBProbA¯+ProbBProbA=Pm1PrPm1Pr+PmPr=Pm1PrPmPmPr+PmPr=Pm1PrPm=1Pr=ProbA¯

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:

ProbB¯/A=ProbA/B¯ProbB¯ProbA=ProbAProbB¯ProbA=PriPrPr=iPr=iPm=ProbB¯=iProbA/B¯ProbB¯ProbA/B¯ProbB¯+ProbA/BProbB=iProbAProbB¯ProbAProbB¯+ProbAProbB=iPriPrPriPr+Pri1Pr=iPriPriPr2+iPriPr2=iPriPriPr=iPr=iPm=ProbB¯

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=Pr+Pm=Pr+i1Pr then:

ProbA/B+ProbB/A=ProbA+ProbB=Pr+Pm=z1AndProbA/B¯+ProbB/A¯=ProbA+ProbB=Pr+Pm=z1ProbA¯/B¯+ProbB¯/A¯=ProbA¯+ProbB¯=1Pr+iPm=z2AndProbA¯/B+ProbB¯/A=ProbA¯+ProbB¯=1Pr+iPm=z2

Therefore, the resultant complex random vector in CPP is:

Z=j=12zj=z1+z2=Pr+1Pr+Pm+iPm=1+i=1+N1i, where N = 2 corresponds to the binary random variable considered in this case. And,

ZN=j=12zjN=z1+z2N=1+N1iN=1N+11Ni=PrZ+PmZ=0.5+0.5i for N = 2 in this case. Thus,

PcZ=PrZ+PmZi=0.5+0.5ii=0.5+0.5=1, just as predicted by CPP.

PrZ=PmZ/i=0.5

ProbZ/NinR=ProbZ/NinM/i=0.5.

To interpret the results obtained, that means that the two probabilities setsR 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=R+M.

SinceC=R+M and Pc2=Pr+Pm/i2=1=Pc in CPP then:

ProbA/B+ProbB/A/i=ProbA+ProbB/i=Pr+Pm/i=1=Pcz1ProbA/B¯+ProbB/A¯/i=ProbA+ProbB/i=Pr+Pm/i=1=Pcz1ProbA¯/B¯+ProbB¯/A¯/i=ProbA¯+ProbB¯/i=1Pr+iPm/i=1=Pcz2ProbA¯/B+ProbB¯/A/i=ProbA¯+ProbB¯/i=1Pr+iPm/i=1=Pcz2

That means that the probability in the set C=R+M is equal to 1, just as predicted by CPP (Table 1).

Probability SetsEvent ProbabilityComplementary Event Probability
In RProbA=PrProbA¯=1ProbA=1Pr
In MProbB=Pm=i1PrProbB¯=iPm=iProbA=iPr
In C=R+Mz1=ProbA+ProbB=Pr+Pmz2=ProbA¯+ProbB¯=1Pr+iPm
Deterministic Probabilities in CPcz1=ProbA+ProbB/i=Pr+Pm/i=1Pcz2=ProbA¯+ProbB¯/i=1Pr+iPm/i=1

### Table 1.

The table of the probabilities inR,M, andC.

#### 4.1.3 The probabilities of dependent and of joint events in C=R+M

ProbAB=ProbAProbB/A=ProbAProbB=ProbBProbA/B=ProbBProbA=PrPm=PmPr=iPr1Pr

And,

ProbAB=ProbA+ProbBProbAB=Pr+PmPrPmProbAB=Pr+i1PrPri1Pr=Pr+iiPriPr+iPr2=Pr+i2iPr+iPr2=Pr+i12Pr+Pr2=Pr+i1Pr2

So, if Pr=1A=R and A¯=andB= and B¯=MProbAB=1=Pr=ProbR, that means we have a 100% deterministic certain experiment A in R.

And if Pr=0A= and A¯=R and B=M and B¯=ProbAB=i=ProbM, that means we have a 100% deterministic impossible experiment A in R.

Moreover,

ProbA¯B=ProbA¯ProbB/A¯=ProbA¯ProbB=1Pr×i1Pr=ProbBProbA¯/B=ProbBProbA¯=i1Pr×1Pr=i1Pr2

And,

ProbA¯B=ProbA¯+ProbBProbA¯B=1Pr+Pmi1Pr2ProbA¯B=1Pr+i1Pri1Pr2=1Pr1+ii1Pr=1Pr1+iPr

So, if Pr=1A=R and A¯=andB= and B¯=MProbA¯B=Prob=0, that means we have a 100% deterministic certain experiment A in R.

And if Pr=0A= and A¯=R and B=M and B¯=.

ProbA¯B=ProbRM=ProbC=1, that means we have a 100% deterministic impossible experiment A in R.

ProbAB¯=ProbAProbB¯/A=ProbAProbB¯=Pr×iPr=ProbB¯ProbA/B¯=ProbB¯ProbA=iPr×Pr=iPr2

And,

ProbAB¯=ProbA+ProbB¯ProbAB¯=Pr+iPriPr2=Pr1+i1Pr

So, if Pr=1A=R and A¯=andB= and B¯=M.

ProbAB¯=ProbRM=ProbC=1, that means we have a 100% deterministic certain experiment A in R.

And if Pr=0A= and A¯=R and B=M and B¯=ProbAB¯=Prob=0, that means we have a 100% deterministic impossible experiment A in R.

Furthermore,

ProbA¯B¯=ProbA¯ProbB¯/A¯=ProbA¯ProbB¯=1Pr×iPr=Pr×i1Pr=ProbB¯ProbA¯/B¯=ProbB¯ProbA¯=iPr×1Pr=Pr×i1Pr=PrPm=PmPr=iPr1Pr

And,

ProbA¯B¯=ProbA¯+ProbB¯ProbA¯B¯=1Pr+iPmPrPm=1Pr+iPrPrPmProbA¯B¯=1Pr+iPrPri1Pr=1Pr+iPriPr+iPr2=1Pr+iPr2

So, if Pr=1A=R and A¯=andB= and B¯=MProbA¯B¯=i=ProbM, that means we have a 100% deterministic certain experiment A in R.

And if Pr=0A= and A¯=R and B=M and B¯=ProbA¯B¯=1=ProbR, that means we have a 100% deterministic impossible experiment A in R (Table 2).

Sets and EventsSets IntersectionSets Union
A, BProbAB=PrPmProbAB=Pr+PmPrPm=Pr+i1Pr2
A¯, BProbA¯B=i1Pr2ProbA¯B=1Pr1+iPr
A,B¯ProbAB¯=iPr2ProbAB¯=Pr1+i1Pr
A¯, B¯ProbA¯B¯=PrPmProbA¯B¯=1Pr+iPrPrPm=1Pr+iPr2

### Table 2.

The table of the probabilities of dependent and of joint events in C=R+M.

Finally, we can directly notice that:

ProbAB=ProbA¯B¯=ProbAProbB=ProbA¯ProbB¯=PrPm=PmPr=iPr1Pr

#### 4.1.4 The relations to CPP parameters

The complex random vector z1=Pr+Pm.

The complex random vector z2=1Pr+iPm.

Therefore, the resultant complex random vector is:

Z=j=12zj=z1+z2=1+i=1+21i=1+N1i, where N = 2 corresponds to the binary random variable that we have studied in this case. Thus,

ZN=PrZ+PmZ=1N+11Ni=12+112i=0.5+0.5iPrZ=0.5andPmZ=0.5i

The Degree of our knowledge or DOKz1 of z1 is: DOKz1=z12=Pr2+Pm/i2.

The Degree of our knowledge or DOKz2 of z2 is: DOKz2=z22=1Pr2+iPm/i2.

The Degree of our knowledge or DOKZ of ZN is:

DOKZ=Z2N2=Z222=1+i24=12+124=PrZ2+PmZ/i2=0.52+0.5i/i2=0.25+0.25=0.5

The Chaotic Factor or Chfz1 of z1 is: Chfz1=2iPrPm.

The Chaotic Factor or Chfz2 of z2 is: Chfz2=2i1PriPm.

The Chaotic Factor or ChfZ of ZN is: ChfZ=2iPrZPmZ=2i0.50.5i=0.5.

The Magnitude of the Chaotic Factor or MChfz1 of z1 is: MChfz1=Chfz1=2iPrPm.

The Magnitude of the Chaotic Factor or MChfz2 of z2 is: MChfz2=Chfz2=2i1PriPm.

The Magnitude of the Chaotic Factor or MChfZ of ZN is:

MChfZ=ChfZ=2iPrZPmZ=2i0.50.5i=0.5=0.5

The probability Pcz1 in C=R+M of z1 is:

Pcz12=Pr+Pm/i2=Pr+1Pr2=12=1=Pcz1

The probability Pcz2 in C=R+M of z2 is:

Pcz22=1Pr+iPm/i2=1Pr+iPr/i2=1Pr+Pr2=12=1=Pcz2

The probability PcZ in C=R+M of ZN is:

PcZ2=PrZ+PmZ/i2=0.5+0.5i/i2=12=1=PcZ

It is important to note here that all the results of the calculations done above confirm the predictions made by CPP.

#### 4.1.5 Bayes’ theorem and CPP and the contingency tables

See Tables 37.

IntersectionAA¯Total
BProbAB=ProbAProbB/A=ProbBProbA/BProbA¯B=ProbA¯ProbB/A¯=ProbBProbA¯/BProbB
B¯ProbAB¯=ProbAProbB¯/A=ProbB¯ProbA/B¯ProbA¯B¯=ProbA¯ProbB¯/A¯=ProbB¯ProbA¯/B¯ProbB¯=iProbB
TotaliProbA=ProbB¯iProbA¯=i1ProbA=ProbBi

### Table 3.

The table of Bayes’ theorem and CPP.

Probabilities in RBB¯
AProbA/B=ProbA=PrProbA/B¯=ProbA=Pr
A¯ProbA¯/B=ProbA¯=1PrProbA¯/B¯=ProbA¯=1Pr
Total11

### Table 4.

The table of the real probabilities in R.

Probabilities in MAA¯
BProbB/A=ProbB=PmProbB/A¯=ProbB=Pm
B¯ProbB¯/A=ProbB¯=iPmProbB¯/A¯=ProbB¯=iPm
Totalii

### Table 5.

The table of the imaginary probabilities in M.

Complex probabilities in C=R+MAA¯
Bz1=ProbA/B+ProbB/A=ProbA+ProbB=Pr+Pmz1=ProbA/B¯+ProbB/A¯=ProbA+ProbB=Pr+Pm
B¯z2=ProbA¯/B+ProbB¯/A=ProbA¯+ProbB¯=1Pr+iPmz2=ProbA¯/B¯+ProbB¯/A¯=ProbA¯+ProbB¯=1Pr+iPm
Total = Resultant Complex Random VectorZ=z1+z2=1+iZ=z1+z2=1+i

### Table 6.

The table of the complex probabilities in C=R+M.

Probability Pc in C=R+MAA¯
BProbA/B+ProbB/A/i=1=Pcz1ProbA¯/B+ProbB¯/A/i=1=Pcz2
B¯ProbA/B¯+ProbB/A¯/i=1=Pcz1ProbA¯/B¯+ProbB¯/A¯/i=1=Pcz2

### Table 7.

The table of the deterministic real probabilities in C=R+M.

### 4.2 The case of a general discrete uniform random variable

#### 4.2.1 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.

Aj is an event occurring in the real probabilities set R such that:

ProbAj=Prj=1N,j:1jN

The corresponding associated imaginary complementary event to the event Aj in the probabilities set M is the event Bj such that:

ProbBj=Pmj=i1Prj=i11N,j:1jN

The real complementary event to the event Aj in R is the event A¯j such that:

AjA¯j=A1A2AjAN=R

and AjAk=,jk (pairwise mutually exclusive events)

ProbA¯j=1ProbAj=1Prj=Pmj/i=ProbBj/i=11NProbR=ProbAjA¯j=ProbA1A2AjAN=ProbA1+ProbA2++ProbAj++ProbAN=N×ProbAj=N×1N=1

The imaginary complementary event to the event Bj in M is the event B¯j such that:

BjB¯j=B1B2BjBN=M

and BjBk=,jk (pairwise mutually exclusive events)

ProbB¯j=iProbBj=iPmj=ii1Prj=ii+iPrj=iPrj=iProbAj=iNProbM=ProbBjB¯j=ProbB1B2BjBN=ProbB1+ProbB2++ProbBj++ProbBN=N×ProbBj=N×i11N=iN1

We have also, as derived from CPP that:

ProbAj/Bj=ProbAj=Prj=1N, that means if the event Bj occurs in M then the event Aj, which is its real complementary event, occurs in R.

ProbBj/Aj=ProbBj=Pmj=i11N, that means if the event Aj occurs in R then the event Bj, which is its imaginary complementary event, occurs in M.

ProbA¯j/B¯j=ProbA¯j=1ProbAj=1Prj=11N, that means if the event B¯j occurs in M then the event A¯j, which is its real complementary event, occurs in R.

ProbB¯j/A¯j=ProbB¯j=iProbBj=iPmj=iPrj=iN, that means if the event A¯j occurs in R then the event B¯j, which is its imaginary complementary event, occurs in M.

#### 4.2.2 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:

ProbAj/B=ProbB/AjProbAjProbB=ProbB/AjProbAjk=1NProbB/AkProbAk

Therefore, in CPP and hence in C=R+M, we can deduce the new forms of Bayes’ theorem for the case considered as follows:

ProbAj/Bj=ProbBj/AjProbAjProbBj=ProbBjProbAjProbBj=ProbAj=ProbBj/AjProbAjk=1NProbBj/AkProbAk==ProbBjProbAjk=1NProbBjProbAk=ProbBjProbAjProbBjk=1NProbAk=ProbBjProbAjProbBj×N1N=ProbAj=1N,j:1jN

And, its corresponding Bayes’ relation in M is:

ProbBj/Aj=ProbAj/BjProbBjProbAj=ProbAjProbBjProbAj=ProbBj=iN1ProbAj/BjProbBjk=1NProbAj/BkProbBk=iN1ProbAjProbBjk=1NProbAjProbBk=iN1ProbAjProbBjProbAjk=1NProbBk=iN1ProbAjProbBjProbAj×iN1=ProbBj=i1ProbAj=i11N,j:1jN

Similarly,

ProbA¯j/B¯j=ProbB¯j/A¯jProbA¯jProbB¯j=ProbB¯jProbA¯jProbB¯j=ProbA¯j=N1ProbB¯j/A¯jProbA¯jk=1NProbB¯j/A¯kProbA¯k=N1ProbB¯jProbA¯jk=1NProbB¯jProbA¯k=N1ProbB¯jProbA¯jProbB¯jk=1NProbA¯k=N1ProbB¯jProbA¯jProbB¯j×N11N=N1ProbB¯jProbA¯jProbB¯j×N1=ProbA¯j=1ProbAj=11N,j:1jN

And, its corresponding Bayes’ relation in M is:

ProbB¯j/A¯j=ProbA¯j/B¯jProbB¯jProbA¯j=ProbA¯jProbB¯jProbA¯j=ProbB¯j=iProbA¯j/B¯jProbB¯jk=1NProbA¯j/B¯kProbB¯k=iProbA¯jProbB¯jk=1NProbA¯jProbB¯k=iProbA¯jProbB¯jProbA¯jk=1NProbB¯k=iProbA¯jProbB¯jProbA¯j×NiN=iProbA¯jProbB¯jProbA¯j×i=ProbB¯j=iProbBj=ii11N=i1ProbA¯j=i111N=iN,j:1jN

Furthermore,

ProbAj/B¯j=ProbB¯j/AjProbAjProbB¯j=ProbB¯jProbAjProbB¯j=ProbAj=ProbB¯j/AjProbAjk=1NProbB¯j/AkProbAk=ProbB¯jProbAjk=1NProbB¯jProbAk=ProbB¯jProbAjProbB¯jk=1NProbAk=ProbB¯jProbAjProbB¯j×N1N=ProbAj=1N,j:1jN

And, its corresponding Bayes’ relation in M is:

ProbBj/A¯j=ProbA¯j/BjProbBjProbA¯j=ProbA¯jProbBjProbA¯j=ProbBj=iN1ProbA¯j/BjProbBjk=1NProbA¯j/BkProbBk=iN1ProbA¯jProbBjk=1NProbA¯jProbBk=iN1ProbA¯jProbBjProbA¯jk=1NProbBk=iN1ProbA¯jProbBjProbA¯j×iN1=ProbBj=i11N,j:1jN

Moreover,

ProbA¯j/Bj=ProbBj/A¯jProbA¯jProbBj=ProbBjProbA¯jProbBj=ProbA¯j=N1ProbBj/A¯jProbA¯jk=1NProbBj/A¯kProbA¯k=N1ProbBjProbA¯jk=1NProbBjProbA¯k=N1ProbBjProbA¯jProbBjk=1NProbA¯k=N1ProbBjProbA¯jProbBj×N11N=N1ProbBjProbA¯jProbBj×N1=ProbA¯j=11N,j:1jN

And, its corresponding Bayes’ relation in M is:

ProbB¯j/Aj=ProbAj/B¯jProbB¯jProbAj=ProbAjProbB¯jProbAj=ProbB¯j=iProbAj/B¯jProbB¯jk=1NProbAj/B¯kProbB¯k=iProbAjProbB¯jk=1NProbAjProbB¯k=iProbAjProbB¯jProbAjk=1NProbB¯k=iProbAjProbB¯jProbAj×NiN=iProbAjProbB¯jProbAj×i=ProbB¯j=iN,j:1jN

Since the complex random vector in CPP is z=Pr+Pm=Pr+i1Pr then:

ProbAj/Bj+ProbBj/Aj=ProbAj/B¯j+ProbBj/A¯j=ProbAj+ProbBj=Prj+Pmj=1N+i11N=zj,j:1jN
ProbA¯j/B¯j+ProbB¯j/A¯j=ProbA¯j/Bj+ProbB¯j/Aj=ProbA¯j+ProbB¯j=Prj+Pmj=11N+iN=zj,j:1jN

Therefore, the resultant complex random vectors in CPP of the uniform discrete random distribution are:

ZU=j=1Nzj=z1+z2++zN=Nzj=N1N+i11N=1+N1i
ZU=j=1Nzj=z1+z2++zN=Nzj=N11N+iN=N1+i

And,

ZUN=j=1NzjN=NzjN=zj=1N+11Ni=PrZU+PmZU. Thus,

PcZU=PrZU+PmZUi=1N+11Nii=1N+11N=1, just as predicted by CPP.

Analogously, ZUN=j=1NzjN=NzjN=zj=11N+iN=PrZU+PmZU. Thus,

PcZU=PrZU+PmZUi=11N+iNi=11N+1N=1, just as predicted by CPP.

SinceC=R+M and Pc2=Pr+Pm/i2=1=Pc in CPP then:

ProbAj/Bj+ProbBj/Aj/i=ProbAj+ProbBj/i=Prj+Pmj/i=1N+11Nii=1N+11N=1=Pcj,j:1jN
ProbAj/B¯j+ProbBj/A¯j/i=ProbAj+ProbBj/i=Prj+Pmj/i=1N+11Nii=1N+11N=1=Pcj,j:1jN
ProbA¯j/B¯j+ProbB¯j/A¯j/i=ProbA¯j+ProbB¯j/i=Prj+Pmj/i=1Prj+iPmj/i=11N+iNi=11N+1N=1=Pcj,j:1jN
ProbA¯j/Bj+ProbB¯j/Aj/i=ProbA¯j+ProbB¯j/i=Prj+Pmj/i=1Prj+iPmj/i=11N+iNi=11N+1N=1=Pcj,j:1jN

That means that the probability in the set C=R+M is equal to 1, just as predicted by CPP.

ProbAjBj=ProbAjProbBj/Aj=ProbAjProbBj=ProbBjProbAj/Bj=ProbBjProbAj=PrjPmj=PmjPrj

Moreover, we have:

ProbAjBj=ProbAj+ProbBjProbAjBj=Prj+PmjPrjPmjProbAjBj=Prj+i1PrjPrji1Prj=Prj+iiPrjiPrj+iPrj2=Prj+i2iPrj+iPrj2=Prj+i12Prj+Prj2=Prj+i1Prj2

So, if Prj=1 then ProbAjBj=Prj=1=ProbR, that means we have a 100% deterministic certain experiment Aj in R.

And if Prj=0 then ProbAjBj=i, that means we have a 100% deterministic impossible experiment Aj in R.

#### 4.2.3 The relations to CPP parameters

The first complex random vector is: zj=Prj+Pmj=1N+11Ni,j:1jN.

Therefore, the first resultant complex random vector is:

ZU=j=1Nzj=z1+z2++zN=Nzj=N1N+11Ni=1+N1i

And, ZUN=PrZU+PmZU=j=1NzjN=NzjN=zj=1N+11Ni.

The second complex random vector is: zj=Prj+Pmj=11N+iN,j:1jN.

Therefore, the second resultant complex random vector is:

ZU=j=1Nzj=z1+z2++zN=Nzj=N11N+iN=N1+i

And, ZUN=PrZU+PmZU=j=1NzjN=NzjN=zj=11N+iN.

The Degree of our knowledge or DOKzj of zj is:

DOKzj=zj2=Prj2+Pmj/i2=1N2+11N2=1+N12N2,j:1jN

The Degree of our knowledge or DOKzj of zj is:

DOKzj=zj2=Prj2+Pmj/i2=11N2+1N2=1+N12N2,j:1jN

The Degree of our knowledge or DOKZU of ZUN is:

DOKZU=ZU2N2=1+N1i2N2=Pr2ZU+PmZUi2=1N2+11N2=1+N12N2

The Degree of our knowledge or DOKZU of ZUN is:

DOKZU=ZU2N2=N1+i2N2=PrZU2+PmZUi2=11N2+1N2=1+N12N2
DOKzj=DOKzj=DOKZU=DOKZU

The Chaotic Factor or Chfzj of zj is:

Chfzj=2iPrjPmj=2i1Ni11N=2N1N2 since i2=1, j:1jN.

The Chaotic Factor or Chfzj of zj is:

Chfzj=2iPrjPmj=2i11Ni1N=2N1N2 since i2=1, j:1jN.

The Chaotic Factor or ChfZU of ZUN is:

ChfZU=2iPrZUPmZU=2i1Ni11N=2N1N2

The Chaotic Factor or ChfZU of ZUN is:

ChfZU=2iPrZUPmZU=2i11Ni1N=2N1N2
Chfzj=Chfzj=ChfZU=ChfZU

The Magnitude of the Chaotic Factor or MChfzj of zj is:

MChfzj=Chfzj=2N1N2=2N1N2,j:1jN

The Magnitude of the Chaotic Factor or MChfzj of zj is:

MChfzj=Chfzj=2N1N2=2N1N2,j:1jN

The Magnitude of the Chaotic Factor or MChfZU of ZUN is:

MChfZU=ChfZU=2N1N2=2N1N2

The Magnitude of the Chaotic Factor or MChfZU of ZUN is:

MChfZU=ChfZU=2N1N2=2N1N2
MChfzj=MChfzj=MChfZU=MChfZU

The probability Pczj in C=R+M of zj is:

Pczj2=Prj+Pmj/i2=1N+11Nii2=1N+11N2=12=1=Pczj,j:1jN

The probability Pczj in C=R+M of zj is:

Pczj2=Prj+Pmj/i2=11N+iNi2=11N+1N2=12=1=Pczj,j:1jN

The probability PcZUin C=R+M of ZUN is:

Pc2ZU=PrZU+PmZUi2=1N+11Nii2=1N+11N2=12=1=PcZU

The probability PcZU in C=R+M of ZUN is:

PcZU2=PrZU+PmZUi2=11N+iNi2=11N+1N2=12=1=PcZU
Pczj=Pczj=PcZU=PcZU=1

It is important to note here that all the results of the calculations done above confirm the predictions made by CPP.

## 5. 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 Lb and the upper bound Ub:

## 6. 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 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.

### 6.1 Simulation of the discrete binomial probability distribution

The probability density function (PDF) of this discrete stochastic distribution is:

fx=CNxpxqNx=NxpxqNx,forLb=0xUb=N

I have taken the domain for the binomial random variable to be: xLb=0Ub=N=10 and k:1k10we haveΔxk=xkxk1=1, then: x=0,1,2,,10.

Taking in our simulation N=10 and p+q=1, p=q=0.5 then:

The mean of this binomial discrete random distribution is: μ=Np=10×0.5=5.

The standard deviation is: σ=Npq=10×0.5×0.5=2.5=1.58113883.

The median is Md=μ=5.

The mode for this symmetric distribution is = 5 = Md = μ.

The cumulative distribution function (CDF) is:

CDFx=ProbXx=k=0xfkN=k=0xNCkpkqNk=k=0x10Ckpkq10k,x:0xN=10

Note that:

If x=0X=LbCDFx=ProbX0=fX=LbN=CN0p0qN0=qN=0.5100.

If x=N=10X=UbCDFx=ProbXx=k=0x=NCNkpkqNk=p+qN=1N=110=1 by the binomial theorem.

The real probability Prjx is:

Prjx=CDFx=k=0xfkN=k=0xNCkpkqNk=k=0x10Ckpkq10k,x:0xN=10
ProbAj/Bj=ProbAj/B¯j=ProbAj=Prjx=k=0xC10kpkq10k

The imaginary complementary probability Pmjx to Prjx is:

Pmjx=i1Prjx=i1CDFx=i1k=0xfkN
=i1k=0xNCkpkqNk=ik=x+1NNCkpkqNk=ik=x+11010Ckpkq10k,x:0xN=10
ProbBj/Aj=ProbBj/A¯j=ProbBj=Pmjx=ik=x+110C10kpkq10k

The real complementary probability Prjx to Prjx is:

Prjx=1Prjx=Pmjx/i=1CDFx=1k=0xfkN=k=x+1NCNkpkqNk=k=x+110C10kpkq10k,x:0xN=10
ProbA¯j/Bj=ProbA¯j/B¯j=ProbA¯j=Prjx=k=x+110C10kpkq10k

The imaginary complementary probability Pmjx to Pmjx is:

Pmjx=iPmjx=ii1Prjx=iPrjx=iCDFx=ik=0xfkN=ik=0xC10kpkq10k,x:0xN=10
ProbB¯j/Aj=ProbB¯j/A¯j=ProbB¯j=Pmjx=ik=0xC10kpkq10k

The complex probability or random vectors are:

zjx=Prjx+Pmjx=k=0xC10kpkq10k+i1k=0xC10kpkq10k=k=0xC10kpkq10k+ik=x+110C10kpkq10k,x:0xN=10
zjx=Prjx+Pmjx=1Prjx+iPmjx=1Prjx+iPrjx=1k=0xC10kpkq10k+ik=0xC10kpkq10k=k=x+110C10kpkq10k+ik=0xC10kpkq10k,x:0xN=10

The Degree of Our Knowledge of zjx:

DOKjx=zjx2=Prj2x+Pmjx/i2=k=0xCNkpkqNk2+1k=0xCNkpkqNk2=1+2iPrjxPmjx=12Prjx1Prjx=12Prjx+2Prj2x=12k=0xCNkpkqNk+2k=0xCNkpkqNk2=12k=0xC10kpkq10k+2k=0xC10kpkq10k2,x:0xN=10.

DOKjx is equal to 1 when Prjx=PrjLb=0=0 and when Prjx=PrjUb=10=1.

The Degree of Our Knowledge of zjx:

DOKjx=zjx2=Prjx2+Pmjx/i2=1Prjx2+iPmjxi2=1k=0xCNkpkqNk2+k=0xCNkpkqNk2=1+2iPrjxPmjx=12Prjx1Prjx=12Prjx+2Prj2x=12k=0xCNkpkqNk+2k=0xCNkpkqNk2=12k=0xC10kpkq10k+2k=0xC10kpkq10k2,x:0xN=10=DOKjx.

DOKjx is equal to 1 when Prjx=PrjLb=0=0 and when Prjx=PrjUb=10=1.

The Chaotic Factor of zjx:

Chfjx=2iPrjxPmjx=2Prjx1Prjx=2Prjx+2Prj2x=2k=0xCNkpkqNk+2k=0xCNkpkqNk2=2k=0xC10kpkq10k+2k=0xC10kpkq10k2,x:0xN=10

Chfjx is null when Prjx=PrjLb=0=0 and when Prjx=PrjUb=10=1.

The Chaotic Factor of zjx:

Chfjx=2iPrjxPmjx=2i1PrjxiPmjx=21PrjxPrjx=2Prjx+2Prj2x=2k=0xCNkpkqNk+2k=0xCNkpkqNk2=2k=0xC10kpkq10k+2k=0xC10kpkq10k2,x:0xN=10=Chfjx

Chfjx is null when Prjx=PrjLb=0=0 and when Prjx=PrjUb=10=1.

The Magnitude of the Chaotic Factor of zjx:

MChfjx=Chfjx=2iPrjxPmjx=2Prjx1Prjx=2Prjx2Prj2x=2k=0xCNkpkqNk2k=0xCNkpkqNk2=2k=0xC10kpkq10k2k=0xC10kpkq10k2,x:0xN=10

MChfjx is null when Prjx=PrjLb=0=0 and when Prjx=PrjUb=10=1.

The Magnitude of the Chaotic Factor of zjx:

MChfjx=Chfjx=2iPrjxPmjx=2i1PrjxiPmjx=21PrjxPrjx=2Prjx2Prj2x=2k=0xCNkpkqNk2k=0xCNkpkqNk2=2k=0xC10kpkq10k2k=0xC10kpkq10k2,x:0xN=10=MChfjx

MChfjx is null when Prjx=PrjLb=0=0 and when Prjx=PrjUb=10=1.

At any value of x: x:Lb=0xUb=N=10, the probability expressed in the complex probability set C=R+M is the following:

Pcj2x=Prjx+Pmjx/i2=zjx22iPrjxPmjx=DOKjxChfjx=DOKjx+MChfjx=1

then,

Pcj2x=Prjx+Pmjx/i2=Prjx+1Prjx2=12=1Pcjx=1always.

And

Pcjx2=Prjx+Pmjx/i2=1Prjx+iPmjxi2=zjx22i1PrjxiPmjx=zjx22iPrjxPmjx=DOKjxChfjx=DOKjx+MChfjx=1

then,

Pcjx2=Prjx+Pmjx/i2=1Prjx+iPmjxi2=1Prjx+ii1Prjxi2=1Prjx+iPrjxi2=1Prjx+Prjx2=12=1Pcjx=1always

Hence, the prediction of all the probabilities and of Bayes’ theorem in the universe C=R+M is permanently certain and perfectly deterministic (Figure 3).

#### 6.1.1 The Complex Probability Cubes.

In the first cube (Figure 4), the simulation of DOK and Chf as functions of each other and of the random variable X for the binomial probability distribution can be seen. The thick line in cyan is the projection of the plane Pc2(X) = DOK(X) – Chf(X) = 1 = Pc(X) on the plane X = Lb = lower bound of X = 0. This thick line starts at the point J (DOK = 1, Chf = 0) when X = Lb = 0, reaches the point (DOK = 0.5, Chf = −0.5) when X = 5, and returns at the end to J (DOK = 1, Chf = 0) when X = Ub = upper bound of X = 10. The other curves are the graphs of 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 = −0.5, X = 5). The point L corresponds to (DOK = 1, Chf = 0, X = Ub = 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 Pr(X) in R and its complementary real probability Pm(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 Pc2(X) = Pr(X) + Pm(X)/i = 1 = Pc(X) on the plane X = Lb = lower bound of X = 0. This thick line starts at the point (Pr = 0, Pm/i = 1) and ends at the point (Pr = 1, Pm/i = 0). The red curve represents Pr(X) in the plane Pr(X) = Pm(X)/i in light grey. This curve starts at the point J (Pr = 0, Pm/i = 1, X = Lb = lower bound of X = 0), reaches the point K (Pr = 0.5, Pm/i = 0.5, X = 5), and gets at the end to L (Pr = 1, Pm/i = 0, X = Ub = upper bound of X = 10). The blue curve represents Pm(X)/i in the plane in cyan Pr(X) + Pm(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 Pr(X) = Pm(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=R+M as a function of the real probability Pr(X) = Re(z) in R and of its complementary imaginary probability Pm(X) = i × Im(z) in M, and this in terms of the random variable X for the binomial probability distribution. The red curve represents Pr(X) in the plane Pm(X) = 0 and the blue curve represents Pm(X) in the plane Pr(X) = 0. The green curve represents the complex probability z(X) = Pr(X) + Pm(X) = Re(z) + i × Im(z) in the plane Pr(X) = iPm(X) + 1 or z(X) plane in cyan. The curve of z(X) starts at the point J (Pr = 0, Pm = i, X = Lb = lower bound of X = 0) and ends at the point L (Pr = 1, Pm = 0, X = Ub = upper bound of X = 10). The thick line in cyan is Pr(X = Lb = 0) = iPm(X = Lb = 0) + 1 and it is the projection of the z(X) curve on the complex probability plane whose equation is X = Lb = 0. This projected thick line starts at the point J (Pr = 0, Pm = i, X = Lb = 0) and ends at the point (Pr = 1, Pm = 0, X = Lb = 0). Notice the importance of the point K corresponding to X = 5 and z = 0.5 + 0.5i when Pr = 0.5 and Pm = 0.5i. The three points J, K, L are the same as in Figure 3.

### 6.2 Simulation of the continuous standard Gaussian normal probability distribution

The probability density function (PDF) of this continuous stochastic distribution is:

fx=dCDFxdx=12πexpx22,for<x<

and the cumulative distribution function (CDF) is:

CDFx=ProbXx=xftdt=x12πexpt22dt

The domain for this standard Gaussian normal variable is considered in the simulations to be equal to: xLb=4Ub=4 and I have taken dx=0.01.

In the simulations, the mean of this standard normal random distribution is μ=0.

The variance is σ2=1.

The standard deviation is σ=1.

The median is Md=0.

The mode for this symmetric distribution is = 0 = Md = μ.

The real probability Prjx is:

Prjx=CDFx=x12πexpt22dt=4x12πexpt22dt,x:4x4
ProbAj/Bj=ProbAj/B¯j=ProbAj=Prjx=4x12πexpt22dt

The imaginary complementary probability Pmjx to Prjx is:

Pmjx=i1Prjx=i1CDFx=i1xftdt
=ix+ftdt=ix+12πexpt22dt=ix412πexpt22dt,x:4x4
ProbBj/Aj=ProbBj/A¯j=ProbBj=Pmjx=ix412πexpt22dt

The real complementary probability Prjx to Prjx is:

Prjx=1Prjx=Pmjx/i=1CDFx=1xftdt=x+ftdt=x412πexpt22dt,x:4x4
ProbA¯j/Bj=ProbA¯j/B¯j=ProbA¯j=Prjx=x412πexpt22dt

The imaginary complementary probability Pmjx to Pmjx is:

Pmjx=iPmjx=ii1Prjx=iPrjx=iCDFx=ixftdt=i4xftdt=i4x12πexpt22dt,x:4x4
ProbB¯j/Aj=ProbB¯j/A¯j=ProbB¯j=Pmjx=i4x12πexpt22dt

The complex probability or random vectors are:

zjx=Prjx+Pmjx=4x12πexpt22dt+i14x12πexpt22dt=4x12πexpt22dt+ix412πexpt22dt,x:4x4
zjx=Prjx+Pmjx=1Prjx+iPmjx=14x12πexpt22dt+i4x12πexpt22dt=x412πexpt22dt+i4x12πexpt22dt,x:4x4

The Degree of Our Knowledge of zjx:

DOKjx=zjx2=Prj2x+Pmjx/i2=4x12πexpt22dt2+14x12πexpt22dt2=1+2iPrjxPmjx=12Prjx1Prjx=12Prjx+2Prj2x=124x12πexpt22dt+24x12πexpt22dt2,x:4x4.

DOKjx is equal to 1 when Prjx=PrjLb=4=0 and when Prjx=PrjUb=4=1.

The Degree of Our Knowledge of zjx:

DOKjx=zjx2=Prjx2+Pmjx/i2=1Prjx2+iPmjxi2=14x12πexpt22dt2+4x12πexpt22dt2=1+2iPrjxPmjx=12Prjx1Prjx=12Prjx+2Prj2x=124x12πexpt22dt+24x12πexpt22dt2,x:4x4=DOKjx.

DOKjx is equal to 1 when Prjx=PrjLb=4=0 and when Prjx=PrjUb=4=1.

The Chaotic Factor of zjx:

Chfjx=2iPrjxPmjx=2Prjx1Prjx=2Prjx+2Prj2x=24xftdt+24xftdt2=24x12πexpt22dt+24x12πexpt22dt2,x:4x4

Chfjx is null when Prjx=PrjLb=4=0 and when Prjx=PrjUb=4=1.

The Chaotic Factor of zjx:

Chfjx=2iPrjxPmjx=2i1PrjxiPmjx=21PrjxPrjx=2Prjx+2Prj2x=24xftdt+24xftdt2=24x12πexpt22dt+24x12πexpt22dt2,x:4x4=Chfjx

Chfjx is null when Prjx=PrjLb=4=0 and when Prjx=PrjUb=4=1.

The Magnitude of the Chaotic Factor of zjx:

MChfjx=Chfjx=2iPrjxPmjx=2Prjx1Prjx=2Prjx2Prj2x=24xftdt24xftdt2=24x12πexpt22dt24x12πexpt22dt2,x:4x4

MChfjx is null when Prjx=PrjLb=4=0 and when Prjx=PrjUb=4=1.

The Magnitude of the Chaotic Factor of zjx:

MChfjx=Chfjx=2iPrjxPmjx=2i1PrjxiPmjx=21PrjxPrjx=2Prjx2Prj2x=24xftdt24xftdt2=24x12πexpt22dt24x12πexpt22dt2,x:4x4=MChfjx

MChfjx is null when Prjx=PrjLb=4=0 and when Prjx=PrjUb=4=1.

At any value of x: x:Lb=4xUb=4, the probability expressed in the complex probability set C=R+M is the following:

Pcj2x=Prjx+Pmjx/i2=zjx22iPrjxPmjx=DOKjxChfjx=DOKjx+MChfjx=1

then,

Pcj2x=Prjx+Pmjx/i2=Prjx+1Prjx2=12=1Pcjx=1always.

And

Pcjx2=Prjx+Pmjx/i2=1Prjx+iPmjxi2=zjx22i1PrjxiPmjx=zjx22iPrjxPmjx=DOKjxChfjx=DOKjx+MChfjx=1

then,

Pcjx2=Prjx+Pmjx/i2=1Prjx+iPmjxi2=1Prjx+ii1Prjxi2=1Prjx+iPrjxi2=1Prjx+Prjx2=12=1Pcjx=1always

Hence, the prediction of all the probabilities and of Bayes’ theorem in the universeC=R+M is permanently certain and perfectly deterministic (Figure 7).

#### 6.2.1 The complex probability cubes

In the first cube (Figure 8), the simulation of DOK and Chf as functions of each other and of the random variable X for the standard Gaussian normal probability distribution can be seen. The thick line in cyan is the projection of the plane Pc2(X) = DOK(X) – Chf(X) = 1 = Pc(X) on the plane X = Lb = lower bound of X = −4. This thick line starts at the point J (DOK = 1, Chf = 0) when X = Lb = −4, reaches the point (DOK = 0.5, Chf = −0.5) when X = 0, and returns at the end to J (DOK = 1, Chf = 0) when X = Ub = upper bound of X = 4. The other curves are the graphs of 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 = −0.5, X = 0). The point L corresponds to (DOK = 1, Chf = 0, X = Ub = 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 Pr(X) in R and its complementary real probability Pm(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 Pc2(X) = Pr(X) + Pm(X)/i = 1 = Pc(X) on the plane X = Lb = lower bound of X = −4. This thick line starts at the point (Pr = 0, Pm/i = 1) and ends at the point (Pr = 1, Pm/i = 0). The red curve represents Pr(X) in the plane Pr(X) = Pm(X)/i in light grey. This curve starts at the point J (Pr = 0, Pm/i = 1, X = Lb = lower bound of X = −4), reaches the point K (Pr = 0.5, Pm/i = 0.5, X = 0), and gets at the end to L (Pr = 1, Pm/i = 0, X = Ub = upper bound of X = 4). The blue curve represents Pm(X)/i in the plane in cyan Pr(X) + Pm(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 Pr(X) = Pm(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=R+M as a function of the real probability Pr(X) = Re(z) in R and of its complementary imaginary probability Pm(X) = i × Im(z) in M, and this in terms of the random variable X for the standard Gaussian normal probability distribution. The red curve represents Pr(X) in the plane Pm(X) = 0 and the blue curve represents Pm(X) in the plane Pr(X) = 0. The green curve represents the complex probability z(X) = Pr(X) + Pm(X) = Re(z) + i × Im(z) in the plane Pr(X) = iPm(X) + 1 or z(X) plane in cyan. The curve of z(X) starts at the point J (Pr = 0, Pm = i, X = Lb = lower bound of X = −4) and ends at the point L (Pr = 1, Pm = 0, X = Ub = upper bound of X = 4). The thick line in cyan is Pr(X = Lb = −4) = iPm(X = Lb = −4) + 1 and it is the projection of the z(X) curve on the complex probability plane whose equation is X = Lb = −4. This projected thick line starts at the point J (Pr = 0, Pm = i, X = Lb = −4) and ends at the point (Pr = 1, Pm = 0, X = Lb = −4). Notice the importance of the point K corresponding to X = 0 and z = 0.5 + 0.5i when Pr = 0.5 and Pm = 0.5i. The three points J, K, L are the same as in Figure 7.

## 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, −0.5 ≤ Chf < 0, and 0 < MChf ≤ 0.5. We can see that during this entire process we have incessantly and continually Pc2 = DOKChf = 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=R+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 Pr, Pm, z and Pc respectively. Consequently, at each value of X, the novel Bayes’ theorem and CPP parameters Pr, Pm, Pm/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 setsR 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 to the random walk problems which have huge and very interesting consequences when implemented to chemistry, to physics, to economics, to applied and pure mathematics.

## Nomenclature

 R real set of events M imaginary set of events C complex set of events i the imaginary number where i=−1 or i2=−1 EKA Extended Kolmogorov’s Axioms CPP Complex Probability Paradigm Prob probability of any event Pr probability in the real set R Pm probability in the imaginary set M corresponding to the real probability in R Pc probability of an event in R with its associated complementary event in M z complex probability number = sum of Pr and Pm = complex random vector DOK = z2 the degree of our knowledge of the random system or experiment, it is the square of the norm of z Chf the chaotic factor of z MChf magnitude of the chaotic factor of z N number of random vectors Z the resultant complex random vector = ∑j=1Nzj DOKZ=Z2N2 the degree of our knowledge of the whole stochastic system ChfZ=ChfN2 the chaotic factor of the whole stochastic system MChfZ magnitude of the chaotic factor of the whole stochastic system ZU the resultant complex random vector corresponding to a uniform random distribution DOKZU the degree of our knowledge of the whole stochastic system corresponding to a uniform random distribution ChfZU the chaotic factor of the whole stochastic system corresponding to a uniform random distribution MChfZU the magnitude of the chaotic factor of the whole stochastic system corresponding to a uniform random distribution PcZU probability in the complex probability set C of the whole stochastic system corresponding to a uniform random distribution

## References

1. 1. Abou Jaoude, A., El-Tawil, K., & Kadry, S. (2010). "Prediction in Complex Dimension Using Kolmogorov’s Set of Axioms", Journal of Mathematics and Statistics, Science Publications, vol. 6(2), pp. 116-124
2. 2. Abou Jaoude, A. (2013)."The Complex Statistics Paradigm and the Law of Large Numbers", Journal of Mathematics and Statistics, Science Publications, vol. 9(4), pp. 289-304
3. 3. Abou Jaoude, A. (2013). "The Theory of Complex Probability and the First Order Reliability Method", Journal of Mathematics and Statistics, Science Publications, vol. 9(4), pp. 310-324
4. 4. Abou Jaoude, A. (2014). "Complex Probability Theory and Prognostic", Journal of Mathematics and Statistics, Science Publications, vol. 10(1), pp. 1-24
5. 5. Abou Jaoude, A. (2015). "The Complex Probability Paradigm and Analytic Linear Prognostic for Vehicle Suspension Systems", American Journal of Engineering and Applied Sciences, Science Publications, vol. 8(1), pp. 147-175
6. 6. Abou Jaoude, A. (2015). “The Paradigm of Complex Probability and the Brownian Motion”, Systems Science and Control Engineering, Taylor and Francis Publishers, vol. 3(1), pp. 478-503
7. 7. Abou Jaoude, A. (2016). “The Paradigm of Complex Probability and Chebyshev’s Inequality”, Systems Science and Control Engineering, Taylor and Francis Publishers, vol. 4(1), pp. 99-137
8. 8. Abou Jaoude, A. (2016). “The Paradigm of Complex Probability and Analytic Nonlinear Prognostic for Vehicle Suspension Systems”, Systems Science and Control Engineering, Taylor and Francis Publishers, vol. 4(1), pp. 99-137
9. 9. Abou Jaoude, A. (2017). “The Paradigm of Complex Probability and Analytic Linear Prognostic for Unburied Petrochemical Pipelines”, Systems Science and Control Engineering, Taylor and Francis Publishers, vol. 5(1), pp. 178-214
10. 10. Abou Jaoude, A. (2017). “The Paradigm of Complex Probability and Claude Shannon’s Information Theory”, Systems Science and Control Engineering, Taylor and Francis Publishers, vol. 5(1), pp. 380-425
11. 11. Abou Jaoude, A. (2017). “The Paradigm of Complex Probability and Analytic Nonlinear Prognostic for Unburied Petrochemical Pipelines”, Systems Science and Control Engineering, Taylor and Francis Publishers, vol. 5(1), pp. 495-534
12. 12. Abou Jaoude, A. (2018). “The Paradigm of Complex Probability and Ludwig Boltzmann's Entropy”, Systems Science and Control Engineering, Taylor and Francis Publishers, vol. 6(1), pp. 108-149
13. 13. Abou Jaoude, A. (2019). “The Paradigm of Complex Probability and Monte Carlo Methods”, Systems Science and Control Engineering, Taylor and Francis Publishers, vol. 7(1), pp. 407-451
14. 14. Abou Jaoude, A. (2020). “Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines and the Complex Probability Paradigm”, Fault Detection, Diagnosis and Prognosis, IntechOpen. DOI: 10.5772/intechopen.90157
15. 15. Abou Jaoude, A. (July 7th 2020). “The Monte Carlo Techniques and The Complex Probability Paradigm”, Forecasting in Mathematics - Recent Advances, New Perspectives and Applications, Abdo Abou Jaoude, IntechOpen. DOI: 10.5772/intechopen.93048
16. 16. Abou Jaoude, A. (2020). “The Paradigm of Complex Probability and Prognostic Using FORM”, London Journal of Research in Science: Natural and Formal (LJRS), London Journals Press, vol. 20(4), pp. 1-65. Print ISSN: 2631-8490, Online ISSN: 2631-8504, DOI: 10.17472/LJRS, 2020
17. 17. Abou Jaoude, A. (2020). “The Paradigm of Complex Probability and The Central Limit Theorem”, London Journal of Research in Science: Natural and Formal (LJRS), London Journals Press, vol. 20(5), pp. 1-57. Print ISSN: 2631-8490, Online ISSN: 2631-8504, DOI: 10.17472/LJRS, 2020
18. 18. Benton, W. (1966). Probability, Encyclopedia Britannica. vol. 18, pp. 570-574, Chicago, Encyclopedia Britannica Inc
19. 19. Benton, W. (1966). Mathematical Probability, Encyclopedia Britannica. vol. 18, pp. 574-579, Chicago, Encyclopedia Britannica Inc
20. 20. Feller, W. (1968). An Introduction to Probability Theory and Its Applications. 3rd Edition. New York, Wiley
21. 21. Walpole, R., Myers, R., Myers, S., & Ye, K. (2002). Probability and Statistics for Engineers and Scientists. 7th Edition, New Jersey, Prentice Hall
22. 22. Freund, J. E. (1973). Introduction to Probability. New York: Dover Publications
23. 23. Abou Jaoude, A. (2019). The Computer Simulation of Monté Carlo Methods and Random Phenomena. United Kingdom: Cambridge Scholars Publishing
24. 24. Abou Jaoude, A. (2019). The Analysis of Selected Algorithms for the Stochastic Paradigm. United Kingdom: Cambridge Scholars Publishing
25. 25. Abou Jaoude, A. (2020). The Analysis of Selected Algorithms for the Statistical Paradigm. The Republic of Moldova: Generis Publishing
26. 26. Abou Jaoude, A. (August 1st 2004). Ph.D. Thesis in Applied Mathematics: Numerical Methods and Algorithms for Applied Mathematicians. Bircham International University. http://www.bircham.edu
27. 27. Abou Jaoude, A. (October 2005). Ph.D. Thesis in Computer Science: Computer Simulation of Monté Carlo Methods and Random Phenomena. Bircham International University. http://www.bircham.edu
28. 28. Abou Jaoude, A. (27 April 2007). Ph.D. Thesis in Applied Statistics and Probability: Analysis and Algorithms for the Statistical and Stochastic Paradigm. Bircham International University. http://www.bircham.edu
29. 29. Stuart, A., Ord, K. (1994). Kendall's Advanced Theory of Statistics: Volume I – Distribution Theory, Edward Arnold, Section 8.7
30. 30. Lee, P. M. (2012). "Chapter 1". Bayesian Statistics. Wiley. ISBN 978-1-1183-3257-3
31. 31. Bayes, T., & Price, R. (1763). "An Essay towards solving a Problem in the Doctrine of Chance. By the late Rev. Mr. Bayes, communicated by Mr. Price, in a letter to John Canton, A. M. F. R. S." (PDF). Philosophical Transactions of the Royal Society of London. 53: pp. 370–418. doi:10.1098/rstl.1763.0053. Archived from the original (PDF) on 2011-04-10. Retrieved 2003-12-27
32. 32. Daston, L. (1988). Classical Probability in the Enlightenment. Princeton University Press. pp. 268. ISBN 0-691-08497-1
33. 33. Stigler, S. M. (1986). "Inverse Probability". The History of Statistics: The Measurement of Uncertainty Before 1900. Harvard University Press. pp. 99–138. ISBN 978-0-674-40341-3
34. 34. Jeffreys, H. (1973). Scientific Inference (3rd edition). Cambridge University Press. pp. 31. ISBN 978-0-521-18078-8
35. 35. Stigler, S. M. (1983). "Who Discovered Bayes' Theorem?". The American Statistician. 37 (4): pp. 290–296. doi:10.1080/00031305.1983.10483122
36. 36. Hooper, M. (2013). "Richard Price, Bayes' theorem, and God". Significance. 10 (1): pp. 36–39. doi:10.1111/j.1740-9713.2013.00638.x. S2CID 153704746
37. 37. Wikipedia, the free encyclopedia, Bayes’ Theorem. https://en.wikipedia.org/

Written By

Abdo Abou Jaoudé

Submitted: 22 April 2021 Reviewed: 11 May 2021 Published: 14 July 2021