## 5 Probabilistic Interpretation of the Results  
The results from the non-linear dynamic analyses in terms of interstory drift index _IDI_ and horizontal relative displacement for the FP devices \(d_{FPS}\) have been useful to perform the probabilistic evaluation of the performance of both FB structure and BI structure. The mentioned above parameters have been probabilistically modeled using a lognormal distribution [17], whose statistical parameters (mean value \(\mu\) and standard deviation \(\sigma\)) are estimated from the numerical results of non-linear dynamic analyses through the maximum likelihood method [40; 41]. The hypothesis of lognormal probabilistic model has been validated trough Chi-square statistical test with level of significance 5%. The monovariate probability density functions (PDFs) have been computed according to:  
\[f(IDI)=\frac{1}{IDI\sigma_{\text{In}(IDI)}\sqrt{2\pi}}\exp\left[-\frac{1}{2} \left(\frac{\ln(IDI)-\mu_{\text{In}(IDI)}}{\sigma_{\text{In}(IDI)}}\right)^{2 }\right] \tag{7}\]  
\[f(d_{FPS})=\frac{1}{d_{FPS}\sigma_{\text{In}(d_{FPS})}\sqrt{2\pi}}\exp\left[- \frac{1}{2}\left(\frac{\ln(d_{FPS})-\mu_{\text{In}(d_{FPS})}}{\sigma_{\text{ In}(d_{FPS})}}\right)^{2}\right] \tag{8}\]  
The probability of exceedance \(P_{f}\) of a given limit state (\(LS\)) has been calculated as follows:  
\[Pf=P(IDI>IDILS)=1-P(IDI\leq IDILS) \tag{9}\]  
where the probability \(P(IDI\leq IDI_{LS})\) can be directly estimated by the lognormal cumulative density functions (CDFs). Figure 12a,b report the monovariate lognormal distributions (PDFs) associated to the _IDIs_ of the different stories with reference to both the FB and BI structure, while Figure 13a,b report the related CDFs with the indication of a generic limit state threshold for evaluation of the associated probability of exceedance \(P_{f}\). Adopting the probabilistic distributions so far described, it is possible to estimate the probability of exceedance \(P_{f}\) for different values of IDIs and \(d_{FPS}\). The results in terms of probability of exceedance \(P_{f}\) are reported in Figure 15a-c with reference to the different stories for the FB and BI structure and for the isolation level. The retrofitting intervention allows to reduce significantly the demand in terms of interstory drift leading to probability of exceedance \(P_{f}\) lower than 1 order of magnitude with respect to the FB structure for specific values of the IDIs. This aspect is fundamental for the safety [42] of the structural systems. Figure 12: Lognormal monovariate PDFs for the _IDIs_ in X and Y direction: FB model (**a**), BI model (**b**). Concerning the BI structure, Figure 14a,b illustrate the monovariate lognormal distributions (PDFs) and the cumulative density functions (CDFs) related to the \(d_{FPS}\) in both the X and Y directions. It can be recognized that the retrofitting intervention is able to reduce the probability of exceedance of pre-determined limit state threshold. Because the structure turns out to be stiffer in the Y direction, both _IDIs_ and \(d_{FPS}\) in that direction are significantly smaller if compared to the values in the X direction. The descending trend of the exceeding probabilities plotted in Figure 15 is perfectly in line with the results described in [3; 17]. In fact, higher values of the limit state thresholds are associated to lower and lower values of \(P_{f}\) in relation to both superstructure and isolation level. The analysis puts in evidence the importance to assess the seismic performance in probabilistic terms for each story and along the both directions for the both models (i.e., FB structure and BI structure). In fact, the results highlight the structural weakness along the X direction and higher values of the exceeding probabilities for lower floors due to also the vertical components of the records. These exceeding probabilities are strongly reduced in the configuration retrofitted through FPS showing a strong plastic reduction of the structural members [18].