**3. Risk identification**

In probability theory, the pair of random variables ð Þ *X*, *Y* is called a twodimensional (bivariate) variable. A nonnegative function *h x*ð Þ , *y* is called the distribution density of a bivariate random variable ð Þ *X*, *Y* , if for any numbers *a*<*b* and

> ð *b*

ð *d*

*h x*ð Þ , *y dydx* (7)

�<sup>∞</sup> *h x*ð Þ , *<sup>y</sup> dx* is the

*c*

� �dy (8)

*a*

2 .

*P*ðf g *ω ϵ* Ω : ðXð Þ *ω ϵ*h i *a*, *b i* ð*Y*ð Þ *ω ϵ*h i *c*, *d* Þ ¼

f xð Þ¼

is the distribution density of the variable *<sup>X</sup>* and *g x*ð Þ¼ <sup>Ð</sup> <sup>þ</sup><sup>∞</sup>

random variable (q, p) as a nonnegative function h(q,p) such that

<sup>2</sup> and Ð <sup>þ</sup><sup>∞</sup>

ðþ<sup>∞</sup> �∞

If instead of classical probability we employ quantum-mechanical density amplitude probability, we arrive at a correct definition. This leads us to conclusively abandon the "objective realism" which determines classical probability and replace it with quantum probability. To examine our reasoning, we shall consider the following example. A pair of random variables (q, p) are given, with known distribution densities. We need to establish the distribution density of the bivariate

�<sup>∞</sup> *h q*ð Þ , *<sup>p</sup> dp* <sup>¼</sup> j j *<sup>ϕ</sup>*ð Þ *<sup>p</sup>*

2.The above relation between *ϕ* and *ψ*, as well as the uncertainty principle, is

3.The expected values of the observable quantum random variables, calculated according to the definition of expected value in probability theory, are equal to the expected values of the same variables, calculated according to quantum

Following Leon Cohen's interpretation, there exist functions fulfilling conditions

Even so, we can treat both *p* and *q* as random variables, but if we consider *p* and *q* jointly, we are leaving probability theory behind: the pair *p*, *q* � � is not a random variable. Instead, we must broaden the applicability of probability theory. None of the above premises contradict our knowledge of real-life phenomena inside road tunnels so it seems justifiable to use quantum probability in risk assessment calculation models. Quantum physics focuses primarily on an object's state which may, for its final manifestation, depend on the tools of analysis we use. The state of the object at a given moment is represented by the direction (radius) in the Hilbert space. A Hilbert space illustrates the type of phase space in which referring to real numbers alone is incomplete. A given state should be treated as a superposition of all possible states at every moment, and quantum mechanics typically deals with complex vector spaces. The resultant model of reality is very rich and composed of many interconnected structures whose states are synchronised and the probability

(1) and (2), but a function fulfilling all three conditions does not exist. Consequently, we cannot go too far in probabilistic interpretations of quantum

of any state's occurrence at any given moment is always the same.

h x, y

*c*<*d* there is equality:

*Risk Management and Assessment*

1.Ð <sup>þ</sup><sup>∞</sup>

mechanics.

**48**

realised.

We can also demonstrate that

distribution density of the variable *Y*.

�<sup>∞</sup> *h q*ð Þ , *<sup>p</sup> dp* <sup>¼</sup> j j *<sup>ψ</sup>*ð Þ*<sup>q</sup>*

operation formalism.

If we want to apply mathematical models to risk analysis, we must clarify our premises and definitions. Risk, in popular understanding, measures the possibility of loss of a given state and may be positive (profit) and negative (loss). Most commonly, "risk" is applied in the context of safety. Most people identify safety as a primary need, without which they experience anxiety and insecurity. It is psychological needs like these that cause individuals, societies, states and organisations to act on their environments in order to remove or reduce factors which increase anxiety, fear, uncertainty or insecurity. As a result, no matter how we define safety, it will ultimately remain an individual interpretation of a given phenomenon. For some people dangerous actions which, if successful, will make them a hero seem right, and their evaluation of possible consequences does not stop them from taking such actions, which would cause fear and inaction in another person. In this context, the security of larger organisations should not rely solely on such subjective assessments. State security is not the same as the sum total of individual securities of each of the state's citizens, and the safety of an organisation is not tantamount to the safety of each of its stakeholders. In the aftermath of the financial crash which bankrupted many companies in 2008, renewed efforts were undertaken to clarify safety for use both in financial management and in other areas of life. In line with the proposed guidelines, safety must be defined as freedom from unacceptable risk. In process safety procedures used in chemical process facilities, "safety" is understood as the absence of unacceptable risk to health, life, property or environment, whereas risk is the product of probability (frequency of occurrence) of a given phenomenon and the scale of losses (size of undesired results) formulated as

$$\text{risk} = \text{probability} \times \text{results} \tag{9}$$

The use of this formula is, however, vitiated by the cognitive determinism of the person identifying the probability and the results. With the 2009 ISO 31000 standard, a new understanding of risk has been proposed. Both the standard and the UN Recommendations on the Transport of Dangerous Goods [6] redefine "risk" as the "effect of uncertainty on objectives". The standard is a collection of frameworks, processes and rules which ought to be complied with during the risk assessment process in every organisation, commercial and otherwise. Building on the earlier considerations, we can reformulate risk as follows:

$$\text{Risk} - \text{[unccertainty]} \text{objective} \tag{10}$$

where "uncertainty" is defined as blurred probability of an event, which cannot be foreseen with absolute certainty, and all the related possibilities and probabilities are variable and possible.

In using measurement instruments (in this context, mathematical analysis) to measure a quantum state, a certain aspect of this state must be adjusted to the state of the instrument used. This is called an observable. In accordance with quantum mechanics'second postulate, each observable is represented by the linear map (vector—Hermitian operator) acting in a Hilbert space, and the eigenvalues of this

operator present all possible results of its measurements. The third postulate proposes that the likelihood, that the measurement of the measurable magnitude of observation A will return a k eigenvalue of the Hermitian operator, equals j j *<sup>λ</sup><sup>k</sup>* <sup>2</sup> , thus confirming the aptness of the risk assessment method, but does not bring us any closer to calculating the actual risk in a particular situation. Real-life events, even in their vector manifestation, do not allow the prediction of the direction of change. It follows that we should limit ourselves to identifying the possibility of each state occurring, based on their probability as derived from probability distribution.

#### **4. Example application**

Risk analyses are typically related to incident analyses. Let us consider the risk of an incident inside a road tunnel. Real data is available: over the last few years, the number of traffic incidents (traffic jams, attempts to reverse, collisions and fires) averages 340 per annum. We can further identify the likely rate of each type of incident within the total number. Let us conduct an analysis of the risk of change to the state of safety, based on the above-discussed postulates.

Graphic representation of uncertainty of incident estimate as shown in **Figure 4**

Conclusion: For the analysed tunnel, there is a 50% uncertainty that the number

of incidents (vehicles stopped) in the year will be between 319 and 375.

3.We must be prepared for no fewer than 319 but ideally 375 incidents.

The analysis allows conclusions which will enable a better preparation for incidents than would have been the case using the standard method, which assumes that the risk of incident equals the product of its value and the probability of its

We ought to understand risk management process as actions coordinated towards the achievement of a predefined state of acceptability. Acceptance of a state, that is to say preparation for it, is the key process in ensuring that an organisation may continue functioning. Risk management process comprises the following

• Risk analysis, or risk magnitude estimation, i.e. quantification of uncertainty in

• Risk recognition, including identification of objectives

• Risk evaluation, i.e. comparison of results with objectives

• Risk identification, including possible consequences

1.We do not know if the number of incidents will equal to 0.

4.We do not know if the number of incidents will exceed 401.

The analysis leads to the following conclusions:

*Application of Quantum Physics Assumptions for Risk Assessment*

*DOI: http://dx.doi.org/10.5772/intechopen.90825*

*Representation of uncertainty of 340 incidents occurring.*

2. It is likely that at least 289 incidents will occur.

above.

**Figure 4.**

occurrence.

stages [7]:

**51**

attaining objectives

**5. Conclusion**

#### **Step 1. Identify the objective**

Accepting the definition of risk as the effect of uncertainty on objectives, we must first identify the objectives. This will be the number of incidents acceptable under the circumstances. Let us assume that the current state is acceptable, which means that we expect 340 incidents in the coming year. Let us take the year to be 365 days. The expected value (objective) will be 381 over the course of the year. Let us now attempt to measure the influence of uncertainty (i.e. the likelihood of a particular distribution) on the objective, i.e. the given annual number of incidents.

#### **Step 2. Identify the event distribution**

Let us assume that incidents occur according to normal distribution. Traffic intensity is constant. There are no planned maintenance works.

For a given tunnel, the number of incidents (vehicles stopped) has been calculated, at certain traffic intensity, as 340 per year.

Calculating the uncertainty of 340 incidents per year (**Table 1**).


#### **Table 1.**

*Uncertainty parameters for the presumed outcome.*

#### **Figure 4.**

operator present all possible results of its measurements. The third postulate proposes that the likelihood, that the measurement of the measurable magnitude of observation A will return a k eigenvalue of the Hermitian operator, equals j j *<sup>λ</sup><sup>k</sup>* <sup>2</sup>

thus confirming the aptness of the risk assessment method, but does not bring us any closer to calculating the actual risk in a particular situation. Real-life events, even in their vector manifestation, do not allow the prediction of the direction of change. It follows that we should limit ourselves to identifying the possibility of each state occurring, based on their probability as derived from probability

Risk analyses are typically related to incident analyses. Let us consider the risk of an incident inside a road tunnel. Real data is available: over the last few years, the number of traffic incidents (traffic jams, attempts to reverse, collisions and fires) averages 340 per annum. We can further identify the likely rate of each type of incident within the total number. Let us conduct an analysis of the risk of change to

Accepting the definition of risk as the effect of uncertainty on objectives, we must first identify the objectives. This will be the number of incidents acceptable under the circumstances. Let us assume that the current state is acceptable, which means that we expect 340 incidents in the coming year. Let us take the year to be 365 days. The expected value (objective) will be 381 over the course of the year. Let us now attempt to measure the influence of uncertainty (i.e. the likelihood of a particular distribution) on the objective, i.e. the given annual number of incidents.

Let us assume that incidents occur according to normal distribution. Traffic intensity is constant. There are no planned maintenance works.

Calculating the uncertainty of 340 incidents per year (**Table 1**).

**Parameter Equation/symbol Value Comments**

3 + 0.02/λ

Mode 340.000 Equals the greatest integer lesser

Skewness √ λ 18.439 Measures the asymmetry of the

340.333

0.0216

than λ

distribution about its mean

For a given tunnel, the number of incidents (vehicles stopped) has been

the state of safety, based on the above-discussed postulates.

distribution.

**4. Example application**

*Risk Management and Assessment*

**Step 1. Identify the objective**

**Step 2. Identify the event distribution**

calculated, at certain traffic intensity, as 340 per year.

Expected mean λ 340.000

Kurtosis 1/λ 0.003 Cumulative probability 0.514

Median *Me* ≈ λ+1/

Probability mass function for the

*Uncertainty parameters for the presumed outcome.*

Poisson distribution

**Table 1.**

**50**

,

*Representation of uncertainty of 340 incidents occurring.*

Graphic representation of uncertainty of incident estimate as shown in **Figure 4** above.

Conclusion: For the analysed tunnel, there is a 50% uncertainty that the number of incidents (vehicles stopped) in the year will be between 319 and 375.

The analysis leads to the following conclusions:


The analysis allows conclusions which will enable a better preparation for incidents than would have been the case using the standard method, which assumes that the risk of incident equals the product of its value and the probability of its occurrence.

### **5. Conclusion**

We ought to understand risk management process as actions coordinated towards the achievement of a predefined state of acceptability. Acceptance of a state, that is to say preparation for it, is the key process in ensuring that an organisation may continue functioning. Risk management process comprises the following stages [7]:


#### *Risk Management and Assessment*

Risk analysis is, in this process, just one of the stages and ought not to be conducted apart from the other elements. Aside from describing dangers and consequences, the process will result in identification of conditions for decision-taking regarding actions which consider the uncertainties of danger and dangerous events occurring, as well as identification of possibilities for avoiding or limiting losses.

**References**

(1903, 1977)

844-849

[1] Heller M. Elementy mechaniki kwantowej dla filozofów. Kraków: Copernicus Center Press; 2017. p. 17

*DOI: http://dx.doi.org/10.5772/intechopen.90825*

*Application of Quantum Physics Assumptions for Risk Assessment*

[2] Hilbert D. Grundlagen der Geometrie. Leipzig: Teubner; 1899

[3] Różycki M. Inertia in procurement risk management. In: Sustainability and Scalability of Business: Theory and Practice. New York: Nova Science Publishers; 2018. pp. 237-245

[4] Schrödinger E. Die gegenwärtige Situation in der Quantenmechanik. Die Naturwissenschaften. 1935;**23**(50):

[5] Wheeler JA. Delayed choice experiments and the Bohr-Einstein dialog. The American Philosophical Society and the Royal Society: papers read at a meeting, June 5, 1980. Physical

Review Letters. 1981;**84**(1):1-5

RID Regulations

html

**53**

[6] Joint ADR, ADN agreement and the

[7] ISO/IEC Guide 51: 2014(en). Safety aspects—Guidelines for their inclusion in standards. Available at: https://www. iso.org/iso-31000-risk-management.

Therefore, the full process ought to comprise calculations of the influence of uncertainty on our objectives and evaluation of results, including the evaluation whether occurrences of motion deviation are counterbalanced by solutions used.

Risk analysis will be most fully realised when we assume that to identify the level of attainment of objectives (i.e. identification of possible risk), we must first identify the probability of each possible state's occurrence, based on probability distribution.

#### **Author details**

Marek Rozycki m/d/r/k Trusted Adviser Group Sp z o.o., Poland

\*Address all correspondence to: m.rozycki@mdrk.eu

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Application of Quantum Physics Assumptions for Risk Assessment DOI: http://dx.doi.org/10.5772/intechopen.90825*
