  For single-$j$ shells with $j={7/2}, {9/2}$ and 11/2, we relate the large
probability of $I^+$ ground states to the largest (smallest) coefficients
$\alpha^J_{I(v \beta)} = <nv \beta I |$ $A^{J \dagger} \cdot A^J | n v\beta
I>$, where $n$ is the particle number, $v$ is the seniority, $\beta$ 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.
