Towards understanding the probability of $0^{+}$ ground states in even-even many-body systems

For single-$j$ shells with $j=7\mathrm{/}2,9\mathrm{/}2$ and 11/2, we relate the large probability of $I^{+}$ ground states to the largest (smallest) coefficients ${\alpha }_{I(v\beta )}^J=<nv\beta I\mathrm{\mid }$ $A^{J \dagger} \cdot A^J | n v\beta I>$,where$n$istheparticlenumber,$v$istheseniority,$\beta$ is an additional quantum number, and $I$ is the angular momentum of the state. Interesting regularities of the probabilities of $I^{+}$ ground states are noticed and discussed for 4-particle systems. Several counter examples of the $0^{+}$ ground state (0GS) predominance are noticed for the first time.