## 4 Application Example
Occurrence probabilities of these specific events determined on the basis of risk criteria [16; 28; 29] and Figure 2 are as presented in Table 2.  
The probability of a peak event related to secondary pollution in the water-supply network was calculated by reference to relationships 1-3, in line with the formulae:  
\[\text{P(a)}=\text{P(b)}\times\text{P(c)}\times\text{P(d)}, \tag{4}\]  
\[\text{P(b)}=\text{P(e)}+\text{P(f)}+\text{P(g)}-\text{P(e)}\times\text{P(f)}- \text{P(f)}\times\text{P(g)}-\text{P(e)}\times\text{P(g)}+\text{P(e)}\times \text{P(f)}\times\text{P(g)}, \tag{5}\]  
\[\text{P(e)}=\text{P(h)}+\text{P(i)}+\text{P(j)}-\text{P(h)}\times\text{P(i)} \cdot\text{P(j)}-\text{P(h)}\times\text{P(j)}+\text{P(h)}\times\text{P(i)} \times\text{P(j)}, \tag{6}\]  
\[\text{P(f)}=\text{P(k)}\times\text{P(l)}\times\text{P(m)}, \tag{7}\]  
In an examined supply network, secondary water pollution has a probability of \(4.3496\times 10^{-8}\). The operational information used to support these issues relates to the operation of a municipal water company-managed supply system in the real world. Its managers' and services' information as well as technical documentation, were used to assess the individual probabilities. Above all, the research discussed here aimed to propose a mechanism for evaluating the risk of changes in water quality in a distribution subsystem. Individual probabilities of secondary water pollution in a supply network calculated based on Equations (8)-(11) are presented in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Probability** & **Description** & **Value** \\ \hline
1. & P(a) & Probability of an event related to the occurrence of secondary water & Calculated acc. to Equation (4) \\
2. & P(b) & Probability of an event related to the occurrence of sediment (growth) & Calculated acc. to Equation (5) \\
3. & P(c) & Probability of an event related to the occurrence of adverse & 0.015 \\
4. & P(d) & Probability of an event related to a sudden change in & 0.0001 \\
5. & P(e) & Probability of an event related to a loss of chemical stability of water & Calculated acc. to Equation (6) \\
6. & P(f) & Probability of an event related to a loss of biological stability of water & Calculated acc. to Equation (7) \\  & & Event probability related to a lack of water physical stability of water & \\
7. & P(g) & resulting in the limit value for water turbidity being exceeded & 0.013 \\  & & (\(\geq\)0.8 NTU) & \\
8. & P(h) & Probability of an event related to the emergence of a water-quality & 0.0001 \\
8. & P(h) & situation with a Langelier Saturation Index value from \(-\)4 to \(-\)5 and from 3 to 4 & 0.0001 \\
9. & P(i) & Event probability related to a loss of chemical stability of water with a & 0.015 \\  & & Ryznar Index value of \(>\)8.5 and \(<\)5.5 & \\
10. & P(j) & Probability of an event related to a lack of chemical stability of water & 0.0002 \\  & & with a Strohecker Index value less than 0.5 & \\
11. & P(k) & Reportedable dissolved organic carbon (BDOC) water-quality value of & 0.583 \\  & & \(\geq\)0.25 g/cm\({}^{3}\) & \\
12. & P(l) & Probability of an event related to a loss of biological stability of water & 0.159 \\
13. & P(m) & Probability of an event related to a lack of water biological stability & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Probabilities of occurrence of specific events. Probabilities were derived on the basis of the relationship:
* Situation under control: P(A) = P(X) \(\cdot\) P(I),
* Failure: P(B) = P(X) \(\times\) (1 \(-\) P(I)) \(\times\) P(II) \(\times\) P(III),
* Serious failure: P(C) = P(X) \(\times\) (1 \(-\) P(I)) \(\times\) P(II) \(\times\) (1 \(-\) P(III)),
* Disaster: P(D) = P(X) \(\times\) (1 \(-\) P(I)) \(\times\) (1 \(-\) P(II)). Probability values for consequences depending on the action of protective barriers are:
* P(A) = 8.699 \(\times\) 10\({}^{-11}\),
* P(B) = 5.209 \(\times\) 10\({}^{-11}\),
* P(C) = 1.250 \(\times\) 10\({}^{-9}\),
* P(D) = 4.211 \(\times\) 10\({}^{-8}\).