In 1933, Andrey Nikolaevich Kolmogorov established the system of five axioms that define the concept of mathematical probability. This system can be developed to include the set of imaginary numbers by adding a supplementary three original axioms. Therefore, any experiment can be performed in the set C of complex probabilities which is the summation of the set R of real probabilities and the set M of imaginary probabilities. The purpose here is to include additional imaginary dimensions to the experiment taking place in the "real" laboratory in R and hence to evaluate all the probabilities. Consequently, the probability in the entire set C = R + M is permanently equal to one no matter what the stochastic distribution of the input random variable in R is; therefore the outcome of the probabilistic experiment in C can be determined perfectly. This is due to the fact that the probability in C is calculated after subtracting from the degree of our knowledge the chaotic factor of the random experiment. Consequently, the purpose in this chapter is to join my complex probability paradigm to the analytic prognostic of buried petrochemical pipelines in the case of linear damage accumulation. Accordingly, after the calculation of the novel prognostic model parameters, we will be able to evaluate the degree of knowledge, the magnitude of the chaotic factor, the complex probability, the probabilities of the system failure and survival, and the probability of the remaining useful lifetime; after that a pressure time t has been applied to the pipeline, which are all functions of the system degradation subject to random and stochastic influences.
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