complex probability

The probability system of five axioms by Andrey Nikolaevich Kolmogorov, proposed in 1933, is extended to take into account the set of imaginary numbers. This is done by adding three new supplementary axioms. As a result, any random experiment can be performed in the extended complex probability set C = R + M, which is the sum of the real set R of real probabilities and the imaginary set M of imaginary probabilities.

The objective is to determine complex probabilities by considering additional imaginary dimensions to the event that occurs in the “real” laboratory. The outcome of the stochastic phenomenon in C can be predicted perfectly, regardless of the probability distribution of the input random variable in R since the corresponding probability in the whole set C is always equal to one. This means that randomness and chance in R are now replaced by absolute determinism in C.

This is because the probability in C is computed after subtracting the chaotic factor from our knowledge of the non-deterministic experiment. This novel complex probability paradigm will be applied to numerical analysis and chaos theory to prove that chaos completely vanishes in the probability universe C = R + M.

Author(s) Details:

Abdo Abou Jaoudé,
Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Notre Dame University-Louaize, Lebanon.

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