Nine friends sit at a round table. Two seatings are the same if everyone has the same neighbours on…

Step 1. All seatings. Fix one friend. The other eight can sit in $8! = 40\,320$ ways.

Step 2. Two side by side. Glue the two into a block. The block and the seven other friends are $8$ objects at a round table, so $7!$ seatings. The two swap inside the block in $2$ ways. So $2 \times 7! = 10\,080$, and $\dfrac{10\,080}{40\,320} = \dfrac{1}{4} = 0.25$. The shortcut $\dfrac{2}{n - 1} = \dfrac{2}{8}$ gives the same.

Step 3. Not side by side. $1 - 0.25 = 0.75$.

Step 4. Three together. The block and six friends are $7$ objects, giving $6! = 720$ seatings. The three swap inside the block in $3! = 6$ ways. So $720 \times 6 = 4320$ favourable seatings, and $\dfrac{4320}{40\,320} = \dfrac{3}{28} \approx 0.1071$.