This is the first one of three companion papers focusing on the "probabilistic evolution approach (PEA)" which has been developed for the solution of the explicit ODE involving problems under certain consistent impositions. The main purpose here is the determination of the expectation value of a given operator in quantum mechanics by solving only ODEs, not directly using the wave function. To this end we first define a basis operator set over the Kronecker powers of an appropriately defined "system operator vector". We assume that the target operator's commutator with the system's Hamiltonian can be expressed in terms of the above-mentioned basis operators. This assumption leads us to an infinite set of linear homogeneous ODEs over the expectation values of the basis operators. Its coefficient matrix is in block Hessenberg form when the target operator has no singularity, and beyond that, it may become block triangular when certain conditions over the system's potential function are satisfied. The initial conditions are the basic determining agents giving the probabilistic nature to the solutions of the obtained infinite set of ODEs. They may or may not have fluctuations depending on the nature of the probability density. All these issues are investigated in a phenomenological and constructive theoretical manner in this paper. The remaining two papers are devoted to further details of PEA in quantum mechanics, and, the application of PEA to systems defined by Liouville equation.