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