We compare the shell-model results for realistic interactions with those obtained for various ensembles of random matrix elements. We show that, although the quantum numbers of the ground states in the even-even nuclei have a high probability ( approximately 60%) to be J(pi)T = 0(+)0, the overlap of those states with the realistic wave functions is very small in average. The transition probabilities B(E2) predicted with random interactions are also too small. The presence of the regular pairing is shown to be a significant element of realistic physics not reproduced by random interactions.