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An original approach to solving rather difficult probabilistic problemsarising in studying the readout of random discrete fields and having no exactanalytical solutions at the moment is proposed. Several algorithms for direct,iterative, and combinatorial-recursive calculations of multidimensionalintegral expressions, which can describe partial solutions of these problems,are presented (these solutions are further used to search for the common closedanalytical regularities). The huge volume of necessary calculations forced usto formalize completely the algorithms and to transfer all the burden ofroutine analytical transforms to a computer. The calculations performed helpedus to establish (and to prove later) a number of new earlier unknownprobabilistic formulas responsible for random division of an interval. One moreimportant feature of this study is the fact that we introduced a new concept of'three-dimensional generalized Catalan numbers' and found their explicit formin solving problems associated with random division of an interval.

The Solving of the Problems with Random Division of an Interval with Use of Computer Analytic Programs