Open access peer-reviewed chapter

The Paradigm of Complex Probability and Thomas Bayes’ Theorem

Written By

Abdo Abou Jaoudé

Submitted: April 22nd, 2021 Reviewed: May 11th, 2021 Published: July 14th, 2021

DOI: 10.5772/intechopen.98340

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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.”

Alfred North Whitehead.

“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.”

Alfred North Whitehead.

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

Bertrand Russell.

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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 Rand hence this will lead to a certain output in the set Cof 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 Rto 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 Rand 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 Call 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.

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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).

Figure 1.

The diagram of the Complex Probability Paradigm major goals.

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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 Ebe a collection of elements {E1, E2, …} called elementary events and let Fbe a set of subsets of Ecalled random events [18, 19, 20, 21, 22]. The five axioms for a finite set Eare:

Axiom 1: Fis a field of sets.

Axiom 2: Fcontains the set E.

Axiom 3: A non-negative real number Prob(A), called the probability of A, is assigned to each set Ain F. We have always 0 ≤ Prob(A) ≤ 1.

Axiom 4: Prob(E) equals 1.

Axiom 5: If Aand Bhave no elements in common, the number assigned to their union is:

ProbAB=ProbA+ProbB

hence, we say that Aand Bare disjoint; otherwise, we have:

ProbAB=ProbA+ProbBProbAB

And we say also that: ProbAB=ProbA×ProbB/A=ProbB×ProbA/Bwhich is the conditional probability. If both Aand Bare independent then: ProbAB=ProbA×ProbB.

Moreover, we can generalize and say that for Ndisjoint (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 Nindependent 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×1Prbe the probability of an associated complementary event in M(the imaginary part) to the event Ain R(the real part). It follows that Pr+Pm/i=1where iis the imaginary number with i=1or i2=1.

Axiom 7: We construct the complex number or vector z=Pr+Pm=Pr+i1Prhaving a norm zsuch that:

z2=Pr2+Pm/i2.

Axiom 8: Let Pcdenote the probability of an event in the complex probability universe Cwhere C=R+M. We say that Pcis the probability of an event Ain Rwith its associated event in Msuch 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 Rof 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 Cof complex numbers which integrates the contributions of both the real set Rof 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+Mbecause we have constantly Pc = 1. In fact, a sure and certain prediction of any random phenomenon is reached in the universe Cbecause 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 Ca probability permanently equal to one as it is shown in the following equation: Pc2 = DOK− Chf = DOK + MChf = 1 = Pcdeduced 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).

Figure 2.

TheEKAor theCPPdiagram.

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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 Aas follows [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]:

Ais an event occurring in the real probabilities set Rsuch that: ProbA=Pr.

The corresponding associated imaginary complementary event to the event Ain the probabilities set Mis the event Bsuch that: ProbB=Pm=i1Pr.

The real complementary event to the event Ain Ris the event A¯such that:

AA¯=Rand 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 Bin Mis the event B¯such that:

BB¯=Mand 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 CPPthat:

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

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

Furthermore, we can deduce from CPPthe following:

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

ProbB¯/A=iPr=iPm=ProbB¯, that means if the event Aoccurs in Rthen 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 Mthen 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 Rthen 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 Requal to:

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

For an epistemological interpretation:

For proposition Aand evidence or background B,

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

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

  • ProbB/Ais the conditional probability or likelihood, the degree of belief in Bgiven that proposition Ais true.

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

  • ProbA/Bis the posterior probability, the probability of Aafter taking into account B.

Therefore, in CPPand 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 Ain Rand correspondingly of Bin M.

And, its corresponding Bayes’ relation in Mis:

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 Ain Rand correspondingly of Bin 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 Ain Rand correspondingly of Bin M.

And, its corresponding Bayes’ relation in Mis:

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 Ain Rand correspondingly of Bin 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 Ain Rand correspondingly of Bin M.

And, its corresponding Bayes’ relation in Mis:

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 Ain Rand correspondingly of Bin 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 Ain Rand correspondingly of Bin M.

And, its corresponding Bayes’ relation in Mis:

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 Ain Rand correspondingly of Bin M.

Since the complex random vector in CPPis z=Pr+Pm=Pr+i1Prthen:

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 CPPis:

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+0.5ifor 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 setsRand Mare 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 Rand Mhave the same chance of 0.5 = 1/2 to be chosen in the complex probabilities set C=R+M.

SinceC=R+Mand Pc2=Pr+Pm/i2=1=Pcin CPPthen:

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+Mis 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

Additionally, we have:

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

And if Pr=0A=and A¯=Rand B=Mand B¯=ProbAB=i=ProbM, that means we have a 100% deterministic impossible experiment Ain 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=Rand A¯=andB=and B¯=MProbA¯B=Prob=0, that means we have a 100% deterministic certain experiment Ain R.

And if Pr=0A=and A¯=Rand B=Mand B¯=.

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

In addition,

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=Rand A¯=andB=and B¯=M.

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

And if Pr=0A=and A¯=Rand B=Mand B¯=ProbAB¯=Prob=0, that means we have a 100% deterministic impossible experiment Ain 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=Rand A¯=andB=and B¯=MProbA¯B¯=i=ProbM, that means we have a 100% deterministic certain experiment Ain R.

And if Pr=0A=and A¯=Rand B=Mand B¯=ProbA¯B¯=1=ProbR, that means we have a 100% deterministic impossible experiment Ain 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 CPPparameters

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 DOKz1of z1is: DOKz1=z12=Pr2+Pm/i2.

The Degree of our knowledge or DOKz2of z2is: DOKz2=z22=1Pr2+iPm/i2.

The Degree of our knowledge or DOKZof ZNis:

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 Chfz1of z1is: Chfz1=2iPrPm.

The Chaotic Factor or Chfz2of z2is: Chfz2=2i1PriPm.

The Chaotic Factor or ChfZof ZNis: ChfZ=2iPrZPmZ=2i0.50.5i=0.5.

The Magnitude of the Chaotic Factor or MChfz1of z1is: MChfz1=Chfz1=2iPrPm.

The Magnitude of the Chaotic Factor or MChfz2of z2is: MChfz2=Chfz2=2i1PriPm.

The Magnitude of the Chaotic Factor or MChfZof ZNis:

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

The probability Pcz1in C=R+Mof z1is:

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

The probability Pcz2in C=R+Mof z2is:

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

The probability PcZin C=R+Mof ZNis:

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 CPPand 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 Pcin 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 Rto illustrate the results obtained for the new Bayes’ theorem when related to CPP.

Ajis an event occurring in the real probabilities set Rsuch that:

ProbAj=Prj=1N,j:1jN

The corresponding associated imaginary complementary event to the event Ajin the probabilities set Mis the event Bjsuch that:

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

The real complementary event to the event Ajin Ris the event A¯jsuch 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 Bjin Mis the event B¯jsuch 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 CPPthat:

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

ProbBj/Aj=ProbBj=Pmj=i11N, that means if the event Ajoccurs in Rthen 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¯joccurs in Mthen 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¯joccurs in Rthen the event B¯j, which is its imaginary complementary event, occurs in M.

4.2.2 The relations to Bayes’ theorem

Bayes’ theorem for Ncompeting statements or hypotheses that is, for Nrandom variables, is in the probability set Requal to:

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

Therefore, in CPPand 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 Mis:

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 Mis:

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 Mis:

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 Mis:

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 CPPis z=Pr+Pm=Pr+i1Prthen:

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 CPPof 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+Mand Pc2=Pr+Pm/i2=1=Pcin CPPthen:

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+Mis equal to 1, just as predicted by CPP.

Additionally, we have:

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=1then ProbAjBj=Prj=1=ProbR, that means we have a 100% deterministic certain experiment Ajin R.

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

4.2.3 The relations to CPPparameters

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 DOKzjof zjis:

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

The Degree of our knowledge or DOKzjof zjis:

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

The Degree of our knowledge or DOKZUof ZUNis:

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

The Degree of our knowledge or DOKZUof ZUNis:

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

The Chaotic Factor or Chfzjof zjis:

Chfzj=2iPrjPmj=2i1Ni11N=2N1N2since i2=1, j:1jN.

The Chaotic Factor or Chfzjof zjis:

Chfzj=2iPrjPmj=2i11Ni1N=2N1N2since i2=1, j:1jN.

The Chaotic Factor or ChfZUof ZUNis:

ChfZU=2iPrZUPmZU=2i1Ni11N=2N1N2

The Chaotic Factor or ChfZUof ZUNis:

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

The Magnitude of the Chaotic Factor or MChfzjof zjis:

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

The Magnitude of the Chaotic Factor or MChfzjof zjis:

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

The Magnitude of the Chaotic Factor or MChfZUof ZUNis:

MChfZU=ChfZU=2N1N2=2N1N2

The Magnitude of the Chaotic Factor or MChfZUof ZUNis:

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

The probability Pczjin C=R+Mof zjis:

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

The probability Pczjin C=R+Mof zjis:

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

The probability PcZUin C=R+Mof ZUNis:

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

The probability PcZUin C=R+Mof ZUNis:

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.

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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 Xis between the lower bound Lband the upper bound Ub:

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6. The new paradigm applied to discrete and continuous stochastic distributions

In this section, the simulation of the novel CPPmodel 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 CPPmodel 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=10and k:1k10we haveΔxk=xkxk1=1, then: x=0,1,2,,10.

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

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=1by the binomial theorem.

The real probability Prjxis:

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

The imaginary complementary probability Pmjxto Prjxis:

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 Prjxto Prjxis:

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 Pmjxto Pmjxis:

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.

DOKjxis equal to 1 when Prjx=PrjLb=0=0and 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.

DOKjxis equal to 1 when Prjx=PrjLb=0=0and 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

Chfjxis null when Prjx=PrjLb=0=0and 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

Chfjxis null when Prjx=PrjLb=0=0and 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

MChfjxis null when Prjx=PrjLb=0=0and 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

MChfjxis null when Prjx=PrjLb=0=0and 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+Mis 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+Mis permanently certain and perfectly deterministic (Figure 3).

Figure 3.

The graphs of all theCPPparameters as functions of the random variableXfor this discrete binomial probability distribution.

6.1.1 The Complex Probability Cubes.

In the first cube (Figure 4), the simulation of DOKand Chfas functions of each other and of the random variable Xfor 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.

Figure 4.

The graphs ofDOKand ofChfand ofPcin terms ofXand of each other for this binomial probability distribution.

In the second cube (Figure 5), we can notice the simulation of the real probability Pr(X) in Rand its complementary real probability Pm(X)/iin Ralso in terms of the random variable Xfor 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)/iin 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)/iin 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.

Figure 5.

The graphs ofPrand ofPm/iand ofPcin terms ofXand of each other for this binomial probability distribution.

In the third cube (Figure 6), we can notice the simulation of the complex probability z(X) in C=R+Mas a function of the real probability Pr(X) = Re(z) in Rand of its complementary imaginary probability Pm(X) = i × Im(z) in M, and this in terms of the random variable Xfor 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 + 0.5iwhen Pr = 0.5 and Pm = 0.5i. The three points J, K, L are the same as in Figure 3.