# Bayes Conditional Probability of Fuzzy Damage and Technical Wear of Residential Buildings
## 2 Research Method
### Problem Identification
Each of the fuzzy sets UII, UIII, UIV can therefore be written using the membership function as follows:  
\[\text{UII, III, IV }=(\sum_{\text{j=1}}^{\text{m}}\text{r(u}_{\text{j}})/\text{u}_{ \text{j}})\text{II, III, IV, gdzie j }\rightarrow\text{m}\in[5,\,12] \tag{28}\]  
and when supplementing the output data with the values of their products:  
\[\text{UII}\bullet\text{UIII} = (\sum_{\text{j=1}}^{\text{m}}\text{r(u}_{\text{j}})\bullet\text{ r(u}_{\text{j}})/\text{u}_{\text{j}})\text{II, III} \tag{29}\] \[\text{UII}\bullet\text{UIV} = (\sum_{\text{j=1}}^{\text{m}}\text{r(u}_{\text{j}})\bullet\text{ r(u}_{\text{j}})/\text{u}_{\text{j}})\text{II, IV}\] (30) \[\text{UIII}\bullet\text{UIV} = (\sum_{\text{j=1}}^{\text{m}}\text{r(u}_{\text{j}})\bullet\text{ r(u}_{\text{j}})/\text{u}_{\text{j}})\text{III, IV}\] (31) \[\text{UII}\bullet\text{UIII}\bullet\text{UIV} = (\sum_{\text{j=1}}^{\text{m}}\text{r(u}_{\text{j}})\bullet\text{ r(u}_{\text{j}})\bullet\text{r(u}_{\text{j}})/\text{u}_{\text{j}})\text{II, III, IV} \tag{32}\]  
When using relationship (4), the degrees of memberships of individual damage \(\mu_{\text{uij}}=\text{r(u}_{\text{j}})\) to sets of groups of damage U (28)-(32), and by having data concerning the probabilities of individual damage in sets ZII, ZIII, ZIV (13)-(15), the partial probabilities of the damage were calculated as fuzzy events in the satisfactory, average, and poor technical maintenance conditions of the analyzed residential buildings:  
\[\text{P(UII) }= (\sum_{\text{j=1}}^{\text{m}}\text{p(u}_{\text{j}})\text{r(u}_{ \text{j}}))\text{II} \tag{33}\] \[\text{P(UIII) }= (\sum_{\text{j=1}}^{\text{m}}\text{p(u}_{\text{j}})\text{r(u}_{ \text{j}}))\text{III}\] (34) \[\text{P(UIV) }= (\sum_{\text{j=1}}^{\text{m}}\text{p(u}_{\text{j}})\text{r(u}_{ \text{j}}))\text{IV} \tag{35}\]  
and their products:  
\[\text{P(UII}\bullet\text{UIII) }= \tag{36}\]  
the conditional probability of the technical wear of an element in relation to its damage increases with the deterioration of the maintenance conditions of the building (this is an exceptionally steady increase, even in the case of different building elements);
* the probability of such a conditionally defined fuzzy event indicates the state of the technical wear for which the fuzzy damage occurs with the highest intensity, and it amounts, for the following elements of the tested residential buildings in their average maintenance condition P(ZIII/U), to:
* for foundations: dampness of foundations 0.40
* for basement walls: crack in bricks 0.39
* for solid floors above basements: dampness of floors 0.38
* for structural walls: cracks of plaster 0.40
* for wooden inter-storey floors: weeping on floors 0.44
* for internal stairs: weeping on stairs 0.46
* for roof constructions: delamination of beams 0.35
* for window joinery: mold and rot on windows 0.37
* for inner plasters: scratches on plaster 0.36
* for facades: scratches on plaster 0.37 The above values are therefore a fuzzy value of the probability of the degree of the technical wear, which was determined as an average degree, i.e., within the range of 35-50%--in the case of the occurrence of a fuzzy damage to the building element;
* this probability is characterized by a low standard deviation and a narrow range of the results of various elements within each of the considered building maintenance conditions--satisfactory (0.2622-0.3846), average (0.3510-0.4613) and poor (0.4995-0.6466).