In this paper we introduce a sparse Bayesian kernel multinomial probit regression model for multi-class cancer classification. The relationship between the cancer types and gene expression measurements is explained by an unknown function which belongs to an abstract functional space like the reproducing kernel Hilbert space. We assign a sparse prior for regression parameters and perform variable selection by indexing the covariates of the model with a binary vector. The correlation prior for the binary vector assigned in this paper is able to distinguish models with the same size. The proposed method is successfully tested on one simulated data set and two publicly available real life data sets.