This paper proposes an original approach to solving complicated probability problems (there is no exact analytical solution) that arise in the reading of random point fields. The schemes of direct, iterative, and combinatorial recursive analytical calculation of multidimensional integral expression that describes the particular solutions of such problems (these solutions are used then to find the general closed analytic dependencies) are shown. A huge amount of required computations forced us to formalize all the algorithms and transfer routine analytical calculations to a computer. The calculations helped us to establish (and later prove) new set of previously unknown probabilistic formulas describing the reliability of reading random point images when such a reading is based on multilevel integrators. Thus, we were able to demonstrate the implementation of the idea proposed by John Von Neumann (extremely rare case in scientific practice): researcher meet difficult and unsolvable problem, use the computer to "suggest" him the right answer, then finds a rigorous proof. Another important feature of our study is that we introduced a new concept of "three-dimensional generalized Catalan numbers" and found their explicit form; this knowledge has been effectively used by us in solving problems related to the registration and analysis of random point images.