Merge branch 'main' into 'release'

Release 2.8

See merge request !425

book/_toc.yml

@@ -269,9 +269,8 @@ parts:
         - file: eva/videos.md
           title: EVA videos 
 
-    # START REMOVE-FROM-PUBLISH
     - file: pd/intro.md
-      title: Risk and Reliability
+      title: Risk Analysis
       # Already in Q1:
       # - file: pd/prob-design/overview.md
       #   sections:
@@ -280,40 +279,42 @@ parts:
       #   - file: pd/prob-design/example-river-system.md
       sections:
       - file: pd/risk-analysis/overview.md
+        title: Risk Introduction
         sections:
         - file: pd/risk-analysis/definition.md
         - file: pd/risk-analysis/steps.md
         - file: pd/risk-analysis/risk-curves.md
-      - file: pd/reliability-component/overview.md
-        sections:
-        - file: pd/reliability-component/contamination.ipynb
+      # - file: pd/reliability-component/overview.md
+      #   sections:
+      #   - file: pd/reliability-component/contamination.ipynb
         # - file: reliability-component/case-two-loads.md
         # - file: reliability-component/case-r-s.md
-      - file: pd/reliability-system/overview.md
-        sections:
+      # - file: pd/reliability-system/overview.md
+        # sections:
         # - file: reliability-system/system-series.md
         # - file: reliability-system/system-parallel.md
-        - file: pd/reliability-system/exercise-simple-city.md
+        # - file: pd/reliability-system/exercise-simple-city.md
       - file: pd/risk-evaluation/overview.md
         sections:
         - file: pd/risk-evaluation/decision.md
         - file: pd/risk-evaluation/cost-benefit.md
         - file: pd/risk-evaluation/econ-optimization.md
-        - file: pd/risk-evaluation/example-dike-height.md
+          sections:
+          - file: pd/risk-evaluation/example-dike-height.md
+            title: Optimization Example
         - file: pd/risk-evaluation/safety-standards.md
       - file: pd/exercises/overview.md
         sections:
         - file: pd/exercises/exercise-fn-curve.md
-        - file: pd/exercises/exercise-paint.md
         - file: pd/exercises/exercise-dam.md
-        - file: pd/exercises/exercise-sample-exam.md
-      - file: pd/videos.md
+        - file: pd/exercises/exercise-paint.md
+        # - file: pd/exercises/exercise-sample-exam.md
+      # - file: pd/videos.md
       # - file: pd/notebooks/overview.md
       #   sections:
       #   - file: pd/notebooks/flood-risk/Exercise_Flood_Risk_Render.ipynb
       #   - file: pd/notebooks/contamination/Exercise_Contamination_Render.ipynb
       #   - file: pd/notebooks/traffic/Exercise_Traffic_Render.ipynb
-    # END REMOVE-FROM-PUBLISH
 
 
   - caption: Programming

book/pd/exercises/exercise-dam.md

 # Dam and River
 
-<!--
-```{admonition} MUDE Exam Information
-:class: tip, dropdown
-Questions 1, 3 and 4 are representative for the Q2 MUDE exam. Question 2 is related.
+```{note}
+Only Questions 3 and 4 of this exercise are relevant for the exam material in this chapter. Note, however, that Questions 1 and 2 should be review of material from previous chapters.
 ```
 
-You are tasked to analyse the safety of a dam and the downstream river system. Downstream of the dam there is a dike ring protected by two main dike sections which must be able to contain water released from the reservoir. The dike sections are connected to each other and form a continuous boundary along the same side of the river. Let $P(F_1)=0.01$ and $P(F_2)=0.01$ denote failure probability of dike section 1 and section 2 respectively. There appears to be a correlation between failures of both sections, $\rho_{1,2}=0.9$, and the figure below shows the effect of the correlation coefficient on the joint failure of the two sections. -->
+You are tasked to analyse the safety of a dam and the downstream river system. Downstream of the dam there is a dike ring protected by two main dike sections which must be able to contain water released from the reservoir. The dike sections are connected to each other and form a continuous boundary along the same side of the river. Let $P(F_1)=0.01$ and $P(F_2)=0.01$ denote failure probability of dike section 1 and section 2 respectively. There appears to be a correlation between failures of both sections, $\rho_{1,2}=0.9$, and the figure below shows the effect of the correlation coefficient on the joint failure of the two sections.
 
 
 % commented paragraphs were used for an exam question with a fault tree

book/pd/exercises/exercise-paint.md

 # Paint System
 
+```{note}
+This exercise is optional; it is provided to illustrate some of the concepts in the decision theory and economics-related pages of the Risk Evaluation chapter, which are not part of the exam material.  
+```
+
 In this assignment we will consider a paint system that prevents rust and corrosion on a steel structure. A high quality paint system costs €40 per square meter and has a failure probability of 0.002, which assumes that the old paint system is completely stripped off and the steel surface is cleaned prior to installation, which has an additional cost of €20 per square meter. 
 
 Common cost cutting measures are to use a cheaper paint that costs half as much, but is five times more likely to fail, or to simply sand the old paint rather than stripping it off prior to applying a new paint, which increases the failure probability by 10, but only costs €5 per square meter.

book/pd/intro.md

-(rr_intro2)=
-# Risk and Reliability, Part 2 
+(risk)=
+# Risk Analysis
 
-```{note}
-Before reading further than this page, be sure to review the {ref}`Part 1 Introduction Page on Risk and Reliability <rr_intro>`, which introduced and defined several key concepts.
-```
-
-Whereas Part 1 of Risk and Reliability defined risk and described how probability is used in the design process, this section focuses on the risk analysis and risk evaluation, and is organized around the following chapters: 
-
-- **Risk Analysis** as a process is formally defined and quantitative risk measures are introduced.
-- **Component Reliability** and **System Reliability** briefly introduce approaches for evaluating reliability, or $\text{P}_\text{bad thing}$, in order to carry out probabilistic assessment and design quantitatively. This is the *quantitative analysis* step of a risk analysis.
-- **Risk evaluation** provides simple quantitative tools and a framework for establishing risk-based safety standards and economic risk criteria. This is a key step in the risk analysis process.
+Throughout this book we have focused on a variety of deterministic and probabilistic topics, which probably appeared to be completely unrelated. However, in engineering practice we often need to combine deterministic and probabilistic approaches to design and assess the projects we work on and Risk Analysis concepts are an important way to facilitate this. In particular, concepts in this chapter are focused not so much on evaluating the behavior of a particular system, but rather _evaluating the risk associated with various outcomes_ and, most importantly, providing _a framework with which decisions can be made to improve it._
 
 **Risk and Reliability in Practice**
 
@@ -19,33 +11,26 @@ Despite the focus on flood management, many applications exist in other fields,
 
 
 ```{admonition} MUDE exam information
-:class: tip, dropdown
+:class: tip
 
 Concepts in this section to focus on for the Q2 MUDE exam are:
-- Definitions of risk and steps of a risk analysis
-- Simple system and component reliability (quantitative risk analysis methods)
-- Use of probability to design and assess engineering systems and components
-- Influence of dependence on simple systems and components
-- Decision analysis, cost-benefit analysis and economic optimization (risk evaluation methods)
-- Evaluation and quantification of risks for a system with three different risk metrics: individual, societal and economic
-- Application and derivation of standards for human safety (individual and societal risk)
-- Application and derivation of standards based on economic risk
-
-Although the list is long, the methods are introduced in a simple form and are always applied to simplified systems of engineering problems within Civil and Environmental Engineering and Geosciences.
+- **Risk Definition** and the **Risk Curve** (Section 8.1)
+- Evaluation of risk using various **Safety Standards** (the last page of Section 8.2)
+- In particular, it is essential that you can construct a risk curve and interpret its meaning, as well as assess it using the limit line approach
+
+A few exercises are provided at the end of this chapter, but do not exhaustively illustrate the type of questions that you can expect on the exam. See exam questions from previous years, as well as in-class activities, for additional examples.
+
 ```
 
 ```{admonition} MUDE not-on-the-exam information
-:class: tip, dropdown
-The following concepts or methods are used in this book to illustrate key subjects and examples, but you will *not* be asked to do them on the exam:
-- List from memory the steps of a risk analysis and describe all aspects in detail
-- Set up a decision tree yourself (note that you may be given a tree with values filled in and asked to interpret it)
-- Define a limit-state function yourself and calculate failure probability
-- Schematize system reliability problems (we will give you one)
-- Evaluate risk curves with more than three scenarios
-- Perform complicated cost benefit analyses
-- By now this list should give you a good enough idea of what to (not) expect...
-
-Exam questions are also designed such that specialized knowledge is not needed to solve them; however, you should be able to recognize loads and resistances and series and parallel systems for any simple civil engineering and geoscience application provided on the exam.
+:class: tip
+
+There are several pages included in this chapter that provides useful pre-requisite ot background information and _you are expected to read and understand them;_ however, these pages do _not_ need to be studied intensively for the exam. This includes:
+
+- Steps of a Risk Analysis (second page in Section 8.1)
+- Decision analysis, cost benefit analysis and economic optimization (first 3 pages in Section 8.2)
+- The Paint System example (Section 8.3)
+
 ```
 
 [^dike]: A dike is a structure, typically made of soil, that protects a specific region from flooding by physically holding back water. Usually associated with rivers, such structures are also widely used on the coast, especially in low elevation areas such as the Netherlands. The Dutch word for levee is *dijk,* but English word *dike* is used in this book. Outside of the Netherlands the words *embankment* and *levee* are used.
\ No newline at end of file

book/pd/risk-analysis/definition.md

@@ -90,6 +90,7 @@ Substantial research has also focused on factors that determine the perception o
 :::{list-table} Formal definitions of risk used in social sciences {cite:p}`vlek1996`
 :header-rows: 1
 :name: risk_definitions
+:widths: 10 90
 
 * - No.
   - Definition

book/pd/risk-analysis/overview.md

-# Risk Analysis
+# Introduction to Risk Analysis
 
-This book splits risk into two parts: risk *analysis* and risk *evaluation.* This chapter considers the analysis of risks from a broad perspective, beginning with definitions and key steps of a risk analysis, illustrated quantitatively through an FN curve. The next two chapters explore quantitative analyses (component and system reliability) before returning to risk evaluation concepts of decision theory, cost benefit analysis and safety standards, where the principal focus is on making decisions and answering the question 'how safe is safe enough?'
+This book splits risk into two parts: risk *introduction* and risk *evaluation.* This chapter in particular introduces the _analysis_ of risks from a broad perspective, beginning with definitions and key steps of a risk analysis, illustrated quantitatively through an FN curve. The next chapter covers risk _evaluation_ concepts decision theory, cost benefit analysis and safety standards, where the principal focus is on making decisions and answering the question 'how safe is safe enough?'
+
+```{note}
+The most important part of this chapter is the definition of risk and risk curves.
+```
 
 <!--
 ```{admonition} MUDE exam information

book/pd/risk-analysis/steps.md

 (risk_steps)=
 # Steps in a Risk Analysis
 
+```{note}
+It is important to read and understand the content on this page, however, it does _not_ need to be studied for the exam. 
+```
 
 The previous section made it clear that risk is a function of probabilities and consequences. Risk analysis, therefore, consists of an analysis of probabilities and consequences associated with undesired events in a given system. Alternative terms used in literature are risk assessment and quantitative risk analysis (QRA). 
 
@@ -50,7 +53,7 @@ The probabilities and consequences of the undesired events identified in step 2
 
 In simple cases the probability can be computed directly, but generally the undesired event must broken down into multiple steps or processes to facilitate the computation of probability, which are then recombined using integration or the total probability theorem. Such an approach results in systems of systems, each of which is composed of individual *components* (or elements), each of which is typically evaluated using the limit state concepts defined above. Often this the probability of a limit state being exceeded is referred to as the probability of failure, which of course reflects the component, not necessarily the system. Techniques for computing the probability of failure are covered in the {ref}`rel_comp` and {ref}`rel_sys` Chapters.
 
-Whereas the most critical aspect of the previous step is identifying the most important failure modes, in the quantitative analysis it is critical to precisely define the quantity of interest, as ambiguity can lead to misunderstandings and incorrect assessment of risk. Often such ambiguities have a direct relationship with the conditional terms used inthe probability computations, which imply specific statements about dependence (or independence). Consider the previous example of river level exceeding the dike height $P(h_w>h_{dike})$: the probability can be significantly different if a design lifetime of one year or fifty years is considered. In addition, the time of year may also play a role as different types of floods may occur. If the seasonal probability of flooding is binary and mutually exclusive with wet and dry season each lasting 6 months of the year, the following expression holds (all values computed an a *per year* basis):
+Whereas the most critical aspect of the previous step is identifying the most important failure modes, in the quantitative analysis it is critical to precisely define the quantity of interest, as ambiguity can lead to misunderstandings and incorrect assessment of risk. Often such ambiguities have a direct relationship with the conditional terms used in the probability computations, which imply specific statements about dependence (or independence). Consider the previous example of river level exceeding the dike height $P(h_w>h_{dike})$: the probability can be significantly different if a design lifetime of one year or fifty years is considered. In addition, the time of year may also play a role as different types of floods may occur. If the seasonal probability of flooding is binary and mutually exclusive with wet and dry season each lasting 6 months of the year, the following expression holds (all values computed an a *per year* basis):
 
 $$
 P(h_w>h_{dike}) = 0.5 \cdot P(h_w>h_{dike}|\mathrm{wet}) + 0.5 \cdot P(h_w>h_{dike}|\mathrm{dry})
@@ -60,7 +63,7 @@ Thus, if the risk analysis were only focusing on floods during the wet season, t
 
 ### Consequences
 
-After failure has been defined, consequences of the events are quantified. First, physical effects associated with an undesired event are considered, sucha s heat and/or smoke from a fire, or inflow of water due to dike breach. Depending on the exposure of people or objects to the physical effects, damages, life loss or other impacts may occur. As an example, a specific consequence, $D$, of dike failure during a flood is considered as a sequence of three discrete events:
+After failure has been defined, consequences of the events are quantified. First, physical effects associated with an undesired event are considered, such as heat and/or smoke from a fire, or inflow of water due to dike breach. Depending on the exposure of people or objects to the physical effects, damages, life loss or other impacts may occur. As an example, a specific consequence, $D$, of dike failure during a flood is considered as a sequence of three discrete events:
 
 - The probability that a dike fails, $P(E_{1})$
 - The conditional probability that water flows into the polder given a dike breach $P(E_{2}|E_{1})$
@@ -133,7 +136,7 @@ In the risk evaluation phase a decision is made whether the risk is acceptable o
 
 **Cost Benefit Analysis**: costs and benefits of risk reduction measures are considered.  When a large number of design choices are possible, an **economic optimization** can be applied to select an optimal system design, based on the costs and benefits of risk reduction.
 
-**Safety Standards**: risk is compared with predetermined safety standards to directly determin acceptability. Such standards are typically imposed by government organizations or standards of practice and often focus on loss of life as the primary risk metric.
+**Safety Standards**: risk is compared with predetermined safety standards to directly determine acceptability. Such standards are typically imposed by government organizations or standards of practice and often focus on loss of life as the primary risk metric.
 
 Given the nature of the key question "how safe is safe enough?", several political, psychological and social processes play a role in the evaluation of risk---in other words: risk evaluation is not purely a technical process, but involves many subjective elements and decisions. One difficulty facing regulators is that preferences and risk attitudes within society may diverge and that costs and benefits may not be distributed evenly, and that a single, collective decision has to be based on strongly divergent individual preferences. In practice, this implies that the establishment of collective decision making procedures is inevitably a political process. This ambiguity can also be found in the numerous interpretations of “the” precautionary principle, which is interpreted by some as a decision making criterion that requires proof of harmlessness (a scientific impossibility), whereas it is seen by others as a decision making procedure that puts emphasis on dialogue and stakeholder involvement, (e.g. {cite:t}`jongejan2008`).
 

book/pd/risk-evaluation/cost-benefit.md

 (cost_benefit)=
 # Cost Benefit Analysis
 
+```{note}
+It is important to read and understand the content on this page, however, it does _not_ need to be studied for the exam. 
+```
+
 This section deals with simplified cost benefit analysis for risk reduction interventions in the engineering domain. An important question in evaluating (engineering) projects is whether the benefits outweigh the costs. Cost benefit analysis (CBA) is generally used for appraisal of a wide range of effects of projects or interventions in order to support decision making. The cost benefit analysis starts with defining the system and existing situation. Then, a broad range of effects of the proposed project and intervention can be identified.
 
 {numref}`costs_and_benefits` below shows an overview of effects of the Delta works that were built after the 1953 flood disaster in the Netherlands. The main aim of the delta works was to provide flood protection to the Southwest of the Netherlands. However, other effects included the agricultural benefits to the region (benefits) and the effects on environmental quality in the estuaries in which dams were built (costs or negative effects).

book/pd/risk-evaluation/decision.md

 (decision)=
 # Decision Analysis
 
+```{note}
+It is important to read and understand the content on this page, however, it does _not_ need to be studied for the exam. 
+```
+
 Decision analysis, or decision-making under uncertain conditions is part of everyday life: when choosing to buy a lottery ticket or choosing to take an umbrella during cloudy weather. In contrast to the rather intuitive decision making in everyday matters, a structured analysis of different alternatives with associated risks, costs and benefits is very useful for decisions in (civil) engineering. This chapter offers a very basic introduction into the decision theory with applications to decision problems in the civil engineering domain. Further reference is made to the work by other scholars for more rigorous and detailed treatment of this topic. See, for example {cite:t}`raiffa1961`; {cite:t}`benjamin1970`.
 
 ## Introduction
@@ -33,7 +37,7 @@ Decision tree.
 
 Based on the possible results a choice is made for an action. To be able to assess the different results, a numerical value is assigned to each outcome, $\omega$, which can be used to establish the benefit of each outcome. This number can be a monetary value, a number on an arbitrary scale or utility--as long as the decision maker(s) can establish a consistent ranking of the outcomes with it. In the last two cases the benefit has no absolute value, but only gives the relative value of the different outcomes.
 
-Utility, $u$, is a concept used to rank the possible outcomes according to the preferences of the decision maker, with possible values $0\leq u \leq 1$ . A utility function can be used to characterize the relative utility of various outcomes. The elaborations below are based on the monetary values as a measure for the outcomes and assume a risk neutral decision maker. This is a decision maker who is indifferent between choices with equal expected outcomes, even if one choice is riskier than the other. For example, a risk neutral decision maker would have the same preference for a € 400 pay out, or a 50/50 bet with a coin toss with outcomes of € 0 (head) or € 800 (tail). Utility and risk aversion are further discussed in ater sections.
+Utility, $u$, is a concept used to rank the possible outcomes according to the preferences of the decision maker, with possible values $0\leq u \leq 1$ . A utility function can be used to characterize the relative utility of various outcomes. The elaborations below are based on the monetary values as a measure for the outcomes and assume a risk neutral decision maker. This is a decision maker who is indifferent between choices with equal expected outcomes, even if one choice is riskier than the other. For example, a risk neutral decision maker would have the same preference for a € 400 pay out, or a 50/50 bet with a coin toss with outcomes of € 0 (head) or € 800 (tail). Utility and risk aversion are further discussed in later sections.
 
 ## Decision rules
 
@@ -178,13 +182,15 @@ The previous analysis has shown that the probability of flooding of the excavati
 Without drainage, the risk, defined as the expected value of the loss, is
 
 $$
-\textrm{risk, without drainage = }0.12 \cdot \euro{} 5,000,000 = \euro{} 600,000
+\textrm{risk, without drainage = }0.12 \cdot \text{€ } 5,000,000 = \text{€ } 600,000
 $$
+
 With drainage the risk is:
 
 $$
-\textrm{risk, with drainage = }0.04 · \euro{} 5,000,000 = \euro{} 200,000
+\textrm{risk, with drainage = }0.04 · \text{€ } 5,000,000 = \text{€ } 200,000
 $$
+
 Costs and probabilities can also be shown in the decision tree (see Figure {numref}`example-river-excavation-3`). The expected values of the costs can be calculated for the different actions by adding the present values of the cost of actions and risk:
 - $\textit a_{1}$ : expected value (additional) costs = risk =€ 600,000
 - $\textit a_{2}$ : expected value (additional) costs 

book/pd/risk-evaluation/econ-optimization.md

 (econ_optimization)=
 # Economic Optimization
 
+```{note}
+It is important to read and understand the content on this page, however, it does _not_ need to be studied for the exam. 
+```
+
 The previous sections have focussed on decisions for which the number of actions was limited, e.g. excavation with or without drainage and the associated costs and benefits. However, there are several situations in which the number of actions is unlimited. This occurs when the failure probability level has to be decided for a system that is yet to be designed, with an infinite number of design options. An example of this type of decision problem is the heightening of dikes, as in theory an unlimited amount of values can be chosen for the elevation, e.g. 2m, 5m, 5.1m, 5.11m, 5.1111m 6m etc.
 
 For this situation an economic optimization that takes into account the costs of increasing the safety level and reducing the risks can be applied to derive an optimal level of safety (or the optimal “failure probability”). The economic optimization was developed and applied by {cite:t}`vandantzig1956`, to derive the optimal dike height for South Holland after the 1953 storm surge disaster, as will be further elaborated in the next section.
@@ -10,7 +14,7 @@ $$
     C_{tot} = I + R
 $$
 
-The annual risk, or expected economic dagmage is found by:
+The annual risk, or expected economic damage is found by:
 
 $$
     E(D) = P_f D
@@ -19,7 +23,7 @@ $$
 where:
 
 - $E(D)$ the expected value of the risk [€/year]
-- $P_f$ the failure probablity of the system per year [1/year]
+- $P_f$ the failure probability of the system per year [1/year]
 - $D$ the damage in case of failure [€]
 
 In this approach it is thus assumed that all damages are expressed in monetary terms. Additional criteria for separately considering the loss of human life are included in the next section. 

book/pd/risk-evaluation/example-dike-height.md

 (ex-dike-height)=
-# Optimization Example
+# Economic Optimization Example
 
-Economic optimization is illustrated through the determination of an optimial dike height to protect against flooding.
+Economic optimization is illustrated through the determination of an optimal dike height to protect against flooding.
 
 Before the major floods of 1953, dikes in the Netherlands were not designed for a specified safety level but mainly strengthened based on practical experience. One of the main questions after the disaster was the optimal dike height and the “acceptable” probability of flooding. Van Dantzig was a professor in mathematics and a member of the first Delta  committee. He developed an econometric approach to determine the  optimal  probability of flooding (or protection level) and the corresponding dike height {cite:p}`vandantzig1956`.
 
@@ -15,7 +15,7 @@ $$ (prob_dist_water_levels)
 
 In which:
 - $h$ the water level [m]
-- $A,B$ constants of the exponential ditribution [m]
+- $A,B$ constants of the exponential distribution [m]
 
 Neglecting wave run-up, the probability of failure of the dikes - leading to flooding -  can be approximated by the probability of exceedance of the dike height $h_d$, i.e.
 

book/pd/risk-evaluation/overview.md

@@ -3,6 +3,10 @@
 
 This chapter focuses on the evaluation of risk (Step 4) by applying several different techniques. While the Sections may seem long, the methods are relatively simple and a lot of examples and explanations are provided to illustrate how they can be applied in practice. After this chapter, you will be ready to try all of the practice problems in the Exercise Chapter.
 
+```{note}
+The most important part of this chapter is the last page, which introduces several techniques for assessing risk. The other sections provide background information and examples that are useful for understanding the concepts, but are not essential for the exam.
+```
+
 <!--
 ```{admonition} MUDE Exam Information
 :class: tip, dropdown

book/pd/risk-evaluation/safety-standards.md

 (safety_standards)=
 # Safety Standards
 
+```{note}
+It is important to read and understand the content on this page, however, it does _not_ need to be studied for the exam. 
+```
+
 When answering the question “how safe is safe enough” a merely economic treatment with cost benefit analysis or economic optimization is often not sufficient for activities with risks to people. Therefore, criteria have been developed that focus on risks to human life. This section focuses on safety standards and criteria for evaluating the risk to life. 
 
 Two aspects are typically considered when evaluating and regulating risks to the public: the total or population-wide effects, and the distribution of effects within the affected population. {numref}`risk_evaluation` summarizes these perspectives. The societal perspective is concerned with ‘total effect’ and the effects of large-scale accidents on the society, in terms of economic damages and life loss. The individual perspective is concerned with distributive justice (‘equity’), i.e. the distribution of harm over the population.
@@ -266,12 +270,12 @@ A risk aversion index $k$ has been introduced to account for risk aversion. For
 
 ````{admonition} Expected value and standard deviation for two systems
 
-We consider two sytems
+We consider two systems
 
-1. This sytem has a high failure probability of 0.01 per year and 1 fatality
+1. This system has a high failure probability of 0.01 per year and 1 fatality
 2. The second system has a smaller failure probability of 0.0001 per year but a higher number of 100 fatalities.
 
-For both systems a binomical distribution of the number of fatalities is applied meaning that the number of fatalities in case of failure is exactly known. The expected value and standard deviation of the number of fatalities are found as follows:
+For both systems a binomial distribution of the number of fatalities is applied meaning that the number of fatalities in case of failure is exactly known. The expected value and standard deviation of the number of fatalities are found as follows:
 
 $$
   E(N) = P_fN \;\;\;\; \sigma^2(N) = P_f(1-P_f)N