# Bayes Conditional Probability of Fuzzy Damage and Technical Wear of Residential Buildings
## 2 Research Method  
### Problem Identification  
The research methodology at a level of greater detail was prepared in such a way that allowed the previously prepared qualitative model to be transformed into a quantitative model. Therefore, the diagnosis of the impact of the maintenance of the residential buildings on the amount of their technical wear was carried out using quantitative methods in fuzzy set categories, and also by using the authors' own model that was created in the conditions of fuzziness. The model allowed for the determination of the conditional probabilities of the process of technical wear, and also the set of damage according to both [PERSON] formulas [40; 41; 42] and the combined approach of [PERSON] [3; 4] and [PERSON] [5; 6];  
As mentioned in the introduction, when visually assessing the technical wear of building elements, the symptoms of their destruction are taken into account, i.e., individual damage that can be categorized into the following groups (groups) of damage:  
* UM--mechanical damage to the structure and texture of building elements;
* UW--damage to building elements caused by water penetration and moisture penetration;
* UD--damage resulting from the loss of the original shape of wooden elements;
* UP--damage to wooden elements attacked by biological pests. The purpose of such a conceptual and technical systematization of damage is a comprehensive diagnosis of the extent to which a building element is worn. This assessment, in turn, leads to the implication of stating under what technical conditions--good, satisfactory, average, poor, or bad--the building element was (is) maintained. It is difficult to define a fuzzy set with such a broad meaning as \"average technical condition of maintenance\"using one membership function. In this case, a semantic analysis of the term \"technical wear of a building element\" was used, which was denoted with the symbol of a fuzzy set \"Z\". Let the technical wear of building element Z consist of: mechanical wear of its structure and texture (fuzzy set ZM), its technical wear caused by water penetration and moisture penetration (fuzzy set ZW), technical wear resulting from the loss of its original shape (fuzzy set ZD), and technical wear caused by the attack of biological pests (fuzzy ZP harvest). This sum can then be expressed as follows:  
\[\text{Z}=\text{ZM}\cup\text{ZW}\cup\text{ZD}\cup\text{ZP} \tag{1}\]  
and when assuming the identity of the degree of technical wear and its visual symptom (Z \(\Leftrightarrow\) U)--damage to a building element that is integrated into the above-described damage sets--this expression takes the following form:  
\[\text{U}=\text{UM}\cup\text{UW}\cup\text{UD}\cup\text{UP} \tag{2}\]  
If technical wear was assumed in the observed states with its measure--the degree of wear as a fuzzy set with no crisp membership boundary of \(\{\text{z}\}=Z\)--then the visual image of this wear--global damage to a building element--should be treated as a fuzzy set, the fuzzy events of which are arguments--distinguished types of damage \(\{\text{u}\}=\text{U}\). Therefore, fuzzy random events are fuzzy sets that express the degree of technical wear, for which there is no complete (measurable) certainty of membership to the II, III or IV class of the technical maintenance of an element. The question then arises: what is the probability of an element being worn, which will more or less represent its average maintenance conditions. To put it simply, what is the probability that an element is more or less (approximately) worn? The approach of [PERSON] [3; 4], who defined the probabilities of fuzzy events in the form of real numbers from interval [0, 1], was used in the research. Therefore, the probability of a fuzzy event, which is the technical wear of a building element, which corresponds to satisfactory, average, and poor maintenance conditions, was defined as:  
\[\text{P}(\text{Z})\text{II},\text{ III},\text{ IV}\;=\;\sum_{\text{i}=1}^{\text{m}} \text{p}(\text{z}_{\text{i}})