2Lase ** Place nine balls numbered $1,2, \cdots, 9$ randomly on nine equally spaced points on a circle, with one ball on each point. Let $S$ be the sum of the absolute differences of the numbers on all adjacent balls on the circle. Find the probability of the arrangement that minimizes $S$. (For example: If one arrangement can be rotated or reflected to match another, they are considered the same arrangement)
Parse: Place nine differently numbered small balls on nine equally divided points on a circle, with one ball on each point. This is equivalent to a circular permutation of nine different elements, so there are $8!$ ways to place them. Considering the factor of flipping, the number of essentially different placements is $\frac{8!}{2}=4 \cdot 7!$.

To find the number of placements that minimize $S$: On the circle, from 1 to 9, there are two paths, the major arc and the minor arc. For any one of these paths, let $x_{1}, x_{2}, \cdots, x_{k}$ be the numbers of the balls sequentially arranged on this arc segment, then
$$
\begin{aligned}
& \left|1-x_{1}\right|+\left|x_{1}-x_{2}\right|+\cdots+\left|x_{k}-9\right| \\
\geqslant & \left|\left(1-x_{1}\right)+\left(x_{1}-x_{2}\right)+\cdots+\left(x_{k}-9\right)\right|=|1-9|=8 .
\end{aligned}
$$

The above inequality holds with equality if and only if $1<x_{1}<x_{2}<\cdots<x_{k}<9$, i.e., the numbers of the balls on each arc segment are arranged in increasing order from 1 to 9. Therefore, $S_{\text {min }}=2 \cdot 8=16$.

From the above, when the set of ball numbers $\left\{1, x_{1}, x_{2}, \cdots, x_{k}, 9\right\}$ on each arc segment is determined, the sorting scheme that achieves the minimum value is uniquely determined.

Among the numbers $1,2, \cdots, 9$, excluding 1 and 9, there are 7 remaining ball numbers $2,3, \cdots, 8$. Dividing them into two subsets, the subset with smaller elements has $\mathrm{C}_{7}^{0}+\mathrm{C}_{7}^{1}+\mathrm{C}_{7}^{2}+\mathrm{C}_{7}^{3}=2^{6}$ cases. Each case corresponds to a unique placement on the circle that minimizes the value of $S$, i.e., the number of favorable events is $2^{6}$. Therefore, the required probability is
$$
P=\frac{2^{6}}{4 \cdot 7!}=\frac{1}{315}
$$