The critical properties of the mixed-spin Ising model consisting of spin-1/2 and 3/2 are investigated in the presence of a bimodal random crystal field by using the effective field theory with correlations. The thermal variations of magnetizations are studied in detail to obtain the phase diagrams. It was found that the model exhibits both second- and first-order phase transitions. The first-order phase transition lines always terminate at the isolated critical points. The model also yields one or two compensation temperatures for appropriate values of the random crystal fields. (C) 2013 Elsevier B.V. All rights reserved.