6.4. Sample Exam Questions#

General Questions#

Question 1: Indicate which of the following statements is true (multiple answers possible):

  1. A parallel system gets stronger if there is a strong positive correlation between failure of the elements.

  2. Dependence between elements is only determined by dependence in loads.

  3. In case of a full dependence between two elements \(A\) and \(B\), it holds that \(P(B|A)=1\).

  4. A flood defence system that is a single line of defence composed of multiple dike sections would best be characterized as a series sytem, not as a parallel system.

  5. A prison with multiple walls and fences would best be characterized as a series system, not as a parallel system.

Flood Protection for a Bay#

A flood protection system for an area at a bay consists of a first line of defence (a dam on the coast) and a second line (a dike around the bay) to protect the city against a storm surge that can cause a high water level in the bay. The design team considers two strategies: 1) a single dike around the bay, or 2) a dike around the bay plus a dam (see figure).

../_images/exercise-sample-exam-bay.png

Question 1: from the perspective of flooding in the city, the function of the dam and dike together is best described as a (choose one): a. series system b. parallel system c. a single component d. none of the above

Question 2: how will dependence between the dam and dike change the system failure probability? (choose one) a. increase b. decrease c. no change

Question 3: describe whether or not there is dependence between the dam and dike, and what the source could be.


The damage in case of flood protection system failure is 1 billion \(10^9\) Euros. We compare two different investments in the system. In strategy 1 the system failure probability becomes \(10^{-3}\) per year at a cost of 10 million (\(10 \cdot 10^{6}\)). In strategy 2 the system failure probability becomes \(3 \cdot 10^{-4}\) per year at a cost of 50 million.

Question 4: for which value of the interest rate \(r\) is strategy 2 the most interesting? (You may consider an infinite lifetime of investments).


A full scale risk assessment is made for the system with probabilities \(P(N>10)=10^{-3}\) per year and \(P(N>100)=5 \cdot 10^{-5}\) per year. A so-called limit line is given with values with \(C = 10^{-1}\); and \(\alpha = 2\).

Question 5: determine whether this situation is acceptable. You may answer using a few sentences, calculations or by submitting a sketch, but either way, you must support your answer quantitatively. If you use a sketch, indicate the scale and relevant points on the plot.


The flooding probability is 1/50 per year and damage is 150 million € (\(10^6\)). It is questioned whether additional protection is needed and what would be the best solution between the following:

  • Strategy A: raise the dike

  • Strategy B: raise the dike and build a mangrove forest in front which reduces waves

The table below indicates the costs of the strategies and how much they reduce the initial failure probability. Assume the interest rate is 4%.

Table 6.1 Mitigation Strategy A and B.#

Strategy

Failure probability factor

Costs, million € (\(10^6\))

Do Nothing

1.0

0

A

0.5

10

B

0.2

18

Question 6: what is the most cost-effective or optimal strategy? Explain your answer.

  1. Strategy A

  2. Strategy B

  3. neither A nor B


Consider the individual risk in two areas of the city by the bay, above. The acceptable individual risk is \(10^{-5}\) per year. The conditional probability of death due to flooding is dependent on the water depth and indicated next to the graph. The individual risk level in area A is \(10^{-5}\) per year and the water depth is 3 m. The individual risk level in area B is \(10^{-5}\) per year and the water depth is 2 m.

Question 7: what should be the maximum allowable failure probability of the dike on the bay, \(P_{\text{flooding}}\), to meet the standard for individual risk, \(IR\)?

../_images/exercise-sample-exam-mortality.png

Component Reliability#

The shaded region, \(\Omega\), in the figure below is the failure domain of an unknown object (the shaded zone extends further towards positive infinity for \(x\) and \(y\)). \(p_f\) is the probability of failure of that object, such that:

\[ p_f = \iint_\Omega f(x,y)dxdy \]

Where \(x\) and \(y\) are random variables and \(f(x,y)\) is the joint PDF.

../_images/exercise-sample-exam-failure-domain.png

Question 1: identify a specific real-life object of your choosing (it can be anything!) that can be described by this diagram and failure probability. Describe the object and provide a definition of failure using words only, no equations. Mention whether \(X\) and \(Y\) each acts as a load or a resistance.