## 2 Methodology
### Analysis of Instability around the Contour of an Underground Exacuation Leading to Overbreak
#### 2.2.2 Probabilistic Method  
The limitations of the deterministic method explained above are reflected in the fact that it does not consider the uncertainty of the input parameters. For a long tunnel and a large amount of processed data, it is necessary to address this limitation. With the aim of better quantifying overbreaks, the probabilistic approach is introduced to calculate the probability of wedge failure and quantify the associated risk. In the probabilistic method, instead of using one (deterministic) value for each parameter, a probability distribution is used to describe the range of possible parameter values and the probability of their occurrence. [PERSON] et al. [21] stated that the selection of probability distributions is site- and parameter-specific, and that there is no universally \"best\" distribution of ground properties. Considering the central limit theorem for sample variance, we can assume that as the sample size gets larger (e.g., >30), the distribution of sampling means approaches a normal distribution. Given the large number of samples in this study (i.e., thousands), a normal distribution of random variables was considered throughout. Instability of the contour of underground excavation is defined using the probability of failure variable, Pf. The probabilistic method comprises the following steps:  
1. Identify the input parameters, which are treated as random variables with a normal probability distribution function (PDF);
2. Apply a probabilistic method to obtain appropriate PDFs and statistical parameters of the limit state function (factor of safety);
3. Evaluate the probability of failure (P\({}_{\mathrm{f}}\)) based on the results of the analysis in step 2. After defining the random variables and their distributions, the sampling method is used to determine how the statistical input distributions for the random variables will be evaluated [22; 23]. For a large number of samples, Monte Carlo simulation is appropriate, by which up to 10,000 samples can be generated for each analysis. By successive execution of Monte Carlo (MC) simulations, the safety factor is obtained for each trial. Finally, the probability density function and the probability distribution of the safety factor (FoS) are calculated to determine the probability of failure (P\({}_{\mathrm{f}}\)). In other words, based on the known PDF of the variables, it is possible to determine the probability that the calculated safety factor for wedge failure will be lower than a given value, which in our case is FoS \(\leq\) 1.0. The probability of failure in the analysis of contour instability of tunnel excavation is defined as the ratio of the number of iterations in which the value of the safety factor is less than one and the total number of iterations. This can be represented by the following equation:  
\[\mathrm{P_{f}}=\frac{\mathrm{Number\ of\ unstable\ wedges\ (where\ FoS\ <\ 1)}}{\mathrm{Total\ number\ of\ analysed\ wedges\ (number\ of\ all\ iterations)}} \tag{8}\]  
Finally, it is necessary to evaluate the range of failure probability associated with a particular event, which, in our case, is the occurrence of overbreaks. Depending on the geotechnical and/or mining engineering application, the admissible failure probability value can have a considerable range (e.g., [24; 25]). When evaluating overbreaks, the question is reversed, so that the values of the probability of failure (0 < P\({}_{\mathrm{f}}\) < 1.0) and the probabilistic safety factor around the contour of the excavation (minimum values) were obtained as output data. Although the interpretation of the deterministic analysis is more than clear, with FoS < 1.0 indicating failure, for the probabilistic analysis, P\({}_{\mathrm{f}}\) > 0.07 (upper limit of 7%) is regarded as a critical probability [26], which should be understood as a more than 7% chance of instantaneous wedge failure around the contour of the unsupported excavation. In the synthesis of deterministic and probabilistic methods, the so-called criterion of stability is defined.