For the case that the elements of anm×nmatrix withm<nare independent discrete uniformly distributed random variables with nonnegative realizations statements on the probability that some choice ofmcolumn vectors forms a basis are given. Moreover, the relative frequency of bases in the constraints of transportation problems is studied. Results obtained permit conclusions on the applicability of such programming methods for problems with linear constraints where every quadratic submatrix of highest order is assumed to be regular.