  Approximations made to estimate two-nucleon transfer probabilities in
ground-state to ground-state transitions and physical interpretation of these
probabilities are discussed. Probabilities are often calculated by
approximating both ground states, of the initial nucleus A and of the final
nucleus A\pm 2 by the same quasiparticle vacuum. We analyze two improvements of
this approach. First, the effect of using two different ground states with
average numbers of particles A and A\pm2 is quantified. Second, by using
projection techniques, the role of particle number restoration is analyzed. Our
analysis shows that the improved treatment plays a role close to magicity,
leading to an enhancement of the pair-transfer probability. In mid-shell
regions, part of the error made by approximating the initial and final ground
states by a single vacuum is compensated by projecting onto good particle
number. Surface effects are analyzed by using pairing interactions with a
different volume-to-surface mixing. Finally, a simple expression of the
pair-transfer probability is given in terms of occupation probabilities in the
canonical basis. We show that, in the canonical basis formulation, surface
effects which are visible in the transfer probability are related to the
fragmentation of single-particle occupancies close to the Fermi energy. This
provides a complementary interpretation with respect to the standard
quasiparticle representation where surface effects are generated by the
integrated radial profiles of the contributing wave functions.
