Coherent perfect absorption (CPA) or reflection (CPR) are methods to realize the extreme manipulation on an optical field. We propose a scheme to operate a bistable switch with convertible CPA and/or CPR. Generally, CPA and CPR occur with different input-field phases. For example, CPA is realized when two input probe beams are in phase; instead, CPR is achieved when they are out of phase. In this scheme, a CPA state can be converted to a CPR state by an incoherent field although two input fields are in phase. When we use the incoherent field as a switching field, the CPA (CPR) state is treated as the closed (open) state. As a result, the switching efficiency can theoretically reach a maximum value, i.e., η = 1. In addition, the switch can be operated in the linear regime with a weak input field, and in the nonlinear or bistable regime with a strong input field. Moreover, the efficiency of the bistable switch is sensitively dependent on the input-field intensity. It provides a potential application of this work on sensitive optical detecting.