Under certain situations, a covariate-adjusted response-adaptive (CARA) design often achieves the benefits of both the covariate- and response-adaptive designs. While almost all existing CARA designs use a prognostic score to balance covariates, the calculation of the prognostic score requires the use of response as the outcome, which renders the dependence of the covariate adaptive probabilities on the response. We propose a Bayesian CARA design (termed BCARA) that allows the incorporation of independent covariate and response adaptive probabilities, which has a meaningful theoretical justification. The proposed method can be implemented either continuously or at each interim analysis.