The mathematical verification of the safety of structures can be done by determining the probability of failure or by using safety elements. Observed damages and collapses are usually assessed within the framework of expert reports, which seems reasonable due to the large number of unique structures in the construction industry. However, there should also be an examination of observed safety across all structures. 
Therefore, in this book the collapse frequencies are determined for different types of structures, such as bridges, dams, tunnels, retaining structures and buildings. The collapse frequency, like the failure probability, belongs to stochasticity. Therefore, the observed mean collapse frequencies and the calculated mean failure probabilities are compared. This comparison shows that the collapse frequencies are usually lower than the calculated failure probabilities. In addition, core damage frequencies and probabilities are given to extend the comparison to another technical product.