The random phase transitions of the Blume-Emery-Griffiths (BEG) model for the spin-1 system are investigated on the Bethe lattice and the phase diagrams of the model are obtained. The biquadratic exchange interaction (K) is turned on, i.e. the BEG model, with probability p either attractively (K > 0) or repulsively (K < 0) and turned off, which leads to the BC model, with the probability (1 - p) throughout the Bethe lattice. By taking the bilinear exchange interaction parameter J as a scaling parameter, the effects of the competitions between the reduced crystal fields (D/J), reduced biquadratic exchange interaction parameter (KU) and the reduced temperature (kT/J) for given values of the probability when the coordination number is q=4, i.e. on a square lattice, are studied in detail. (C) 2015 Elsevier B.V. All rights reserved.