As opposed to the reliability, the failure probability is the other side of a coin. To figure out the failure probability of a rotor-bearing dynamic system, the probabilistic method is employed to analyze a Limit-State function which consists of a failure/success criterion and the result from a deterministic analysis. In other words, the sign (negative for failure/positive for success) of the limit-state function indicates the failure/success status of a rotor-bearing dynamic system. Mathematical methods such as First Order Reliability Method (FORM), Second-Order Reliability Method (SORM), Simulation Methods (Monte Carlo and modified versions) and Response Surface Method, can be used to calculate probability distribution function (PDF) and apply integration to obtain cumulative distribution function (CDF) or the final failure probability or reliability.

In the reliability analysis, design parameters are considered as distribution functions which may not be known or available. Generally, some statistical distribution functions are assigned to represent the random characteristics of the design variables.

In addition to the determination of reliability, the significance of the random variables affecting the over all system behavior can be realized. Therefore, the so-called sensibility for design variables can be utilized to select sensible variables and lead to robust design, optimization, of a reliable rotor-bearing system.

​