We characterize the positional social preference correspondences (spc) satisfying the qualified majority property for any given majority threshold. We also characterize the positional spcs satisfying the minimal majority property. We next evaluate the probability that the Borda, the plurality and the antiplurality spcs fulfil the two aforementioned properties under the Impartial and Anonymous Culture assumption in the presence of three and four alternatives for various sizes of the society. Our results show that the Borda spc is the positional spc which (...) better behaves in relation with the qualified majority principle and the minimal majority principle. (shrink)