We analyze the ground state of the two-dimensional quantum system of electrons confined in a parabolic potential with the system size around 100 at 0 K. We map the system onto a classical system on the basis of the classical-map hypernetted-chain (CHNC) method which has been proven to work in the integral-equation-based analyses of uniform systems and apply classical Monte Carlo and molecular dynamics simulations. We find that, when we decrease the strength of confinement keeping the number of confined electrons fixed, the energy of the spin-polarized state with somewhat lower average density becomes smaller than that of the spin-unpolarized state with somewhat higher average density. This system thus undergoes the transition from the spin-unpolarized state to the spin polarized state and the corresponding critical value of r(s) estimated from the average density is as low as r(s) similar to 0.4 which is much smaller than the r(s) value for the Wigner lattice formation. When we compare the energies of spin-unpolarized and spin-polarized states for given average density, our data give the critical r(s) value for the transition between unpolarized and polarized states around 10 which is close to but still smaller than the known possibility of polarization at r(s) similar to 27. The advantage of our method is a direct applicability to geometrically complex systems which are difficult to analyze by integral equations and this is an example. (C) 2008 Elsevier B.V. All rights reserved.