Probabilistic Risk Analysis

We established that a risk analysis answers three questions: (i) What can happen? (ii) How likely is it that that will happen, and (iii) If it does happen what are the consequences? To answer these questions typically a triplet is used \(〈s_i,p_i,x_i 〉\). Where \(s_i\) is a scenario description, \(p_i\) is the probability of the scenario and \(x_i\) is the consequence of the scenario. Risk can be defined as a set of triplets \(R={〈s_i,p_i,x_i 〉}\), \(i=1,…,N\). Because the possible scenarios in any risk analysis are infinite, and in practice the risk analyst cannot contemplate all these, an N+1 scenario that accounts for all “other” possibilities is often used.
The triplets \(〈s_i,p_i,x_i 〉\) are often presented in a tabular form as

Scenario	Probability	Consequence
\(S_1\)	\(p_1\)	\(x_1\)
\(S_2\)	\(p_2\)	\(x_2\)
⋮	⋮	⋮
\(S_N\)	\(p_N\)	\(x_N\)

For example, imagine that you are analyzing the risk of a bridge connecting two sides of a river. We can define five events:

1. Failure of the bridge due to the support beams;

2. Failure due to scouring around the bridge foundation;

3. Stop of the traffic due to a car accident;

4. Stop of the traffic due to damages in the pavement;

5. Normal operation of the bridge.

Once the scenarios are defined, we can define the triplets

Scenario	Probability	Consequence
1	\(0.04\)	\(2 \cdot 10^6\)
2	\(0.03\)	\(3\cdot 10^5\)
3	\(0.10\)	\(5\cdot 10^4\)
4	\(0.08\)	\(7\cdot 10^5\)
5	\(1-0.04-0.03-0.10-0.08 = 0.75\)	\(0\)

where the consequences here are assessed in terms of economic consequences (€).

Risk Curves

To present a visual idea of risk, we first arrange the scenarios in order of severity. That is such that \(x_1 \leq x_2 \leq x_3 \leq ⋯ \leq x_N\). By adding a column to the table above we may compute the cumulative probabilities adding from the bottom.

Scenario	Probability	Consequence	Cumulative Probabililty
\(S_1\)	\(p_1\)	\(x_1\)	\(P_1 = P_2+p_1\)
\(S_2\)	\(p_2\)	\(x_2\)	\(P_2 = P_3+p_2\)
⋮	⋮	⋮	⋮
\(S_{N-1}\)	\(p_{N-1}\)	\(x_{N-1}\)	\(P_{N-1} = P_N+p_{N-1}\)
\(S_N\)	\(p_N\)	\(x_N\)	\(P_N = p_N\)

A plot of the consequences \(x_i\) against the cumulative probabilities \(P_i\) looks like shown in the Figure below.

Fig. 8.1 Example of risk curve.

A common representation is in a log-log scale for both axis in which case the (smoothed risk) curve looks like in the Figure below.

Fig. 8.2 Example of risk curve in log-log scale.

Going back to the previous example of the bridge, we can order the triplets in decreasing order of severity as

Scenario	Probability	Consequence	Cumulative Probabililty
5	\(1-0.04-0.03-0.10-0.08 = 0.75\)	\(0\)	\(1.00\)
3	\(0.10\)	\(5\cdot 10^4\)	\(0.25\)
2	\(0.03\)	\(3\cdot 10^5\)	\(0.15\)
4	\(0.08\)	\(7\cdot 10^5\)	\(0.12\)
1	\(0.04\)	\(2 \cdot 10^6\)	\(0.04\)

Finally, we can represent the risk curve as

Fig. 8.3 Risk curve for the bridge example.

And in log-log scale as

Fig. 8.4 Risk curve in log-log scale for the bridge example.

Attribution

This chapter is written by Patricia Mares Nasarre, Max Ramgraber and Oswaldo Morales Napoles. Find out more here.