**3.1 The original Andrey Nikolaevich Kolmogorov system of axioms**

The simplicity of Kolmogorov's system of axioms may be surprising [4–22]. Let *E* be a collection of elements {*E*1, *E*2, … } called elementary events, and let *F* be a set of subsets of *E* called random events [23–27]. 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 nonnegative 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

$$P\_{rob}(A \cup B) = P\_{rob}(A) + P\_{rob}(B)$$

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

$$P\_{rob}(A \cup B) = P\_{rob}(A) + P\_{rob}(B) - P\_{rob}(A \cap B)$$

And we say also that: *Prob*ð Þ¼ *A* ∩ *B Prob*ð Þ� *A Prob*ð Þ¼ *B=A Prob*ð Þ� *B Prob*ð Þ *A=B* which is the conditional probability. If both *A* and *B* are independent then: *Prob*ð Þ¼ *A* ∩ *B Prob*ð Þ� *A Prob*ð Þ *B* .

Moreover, we can generalize and say that for *N* disjoint (mutually exclusive) events *A*1,*A*2, … ,*Aj*, … ,*AN* (for 1≤*j*≤ *N*), we have the following additivity rule:

$$P\_{rob}\left(\bigcup\_{j=1}^{N}\mathcal{A}\_{j}\right) = \sum\_{j=1}^{N}P\_{mb}\left(\mathcal{A}\_{j}\right)$$

And we say also that for *N* independent events *A*1,*A*2, … ,*Aj*, … ,*AN* (for 1≤*j*≤ *N*), we have the following product rule:

$$P\_{rob}\left(\bigcap\_{j=1}^N A\_j\right) = \prod\_{j=1}^N P\_{mb}\left(A\_j\right)$$

### **3.2 Adding the imaginary part M**

union is:

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

**Axiom 6:** Let *Pm* ¼ *i* � ð Þ 1 � *Pr* be the probability of an associated complementary event in **M** (the imaginary part or universe) to the event *A* in **R** (the real part or universe). It follows that *Pr* þ *Pm=i* ¼ 1, where *i* is the imaginary number with *<sup>i</sup>* <sup>¼</sup> ffiffiffiffiffiffi �<sup>1</sup> <sup>p</sup> or *<sup>i</sup>* <sup>2</sup> ¼ �1.

**Axiom 7:** We construct the complex number or vector *Z* ¼ *Pr* þ *Pm* ¼ *Pr* þ *i*ð Þ 1 � *Pr* having a norm j j *Z* such that:

$$\left| Z \right|^2 = P\_r^2 + \left( P\_m / i \right)^2.$$

**Axiom 8:** Let *Pc* denote the probability of an event in the complex probability set and universe **C**, where **C** ¼ **R** þ**M**. We say that *Pc* is the probability of an event *A* in **R** with its associated and complementary event in **M** such that:

$$\text{Pc}^2 = \left(P\_r + P\_m/i\right)^2 = |Z|^2 - 2\text{i}P\_r P\_m \text{ and is always equal to } \mathbf{1}.$$

We can see that by taking into consideration the set of imaginary probabilities we added three new and original axioms and consequently the system of axioms

**Figure 2.** *The* EKA *or the* CPP *diagram.*

defined by Kolmogorov was hence expanded to encompass the set of imaginary numbers and realm [28–65].
