Seat friends at a dinner table I have the following question:
At a squared table I seat $8$ friends of mine.
At each side, I place two friends.
Always two friends know each other.
What's the probability that no friend knows it's sided, neighbor?
I tried to solve it by:


*

*Calculating all different seating possibilities: $4!$ $\Rightarrow 24$

*And writing down all options on paper where no side neighbors know the other side neighbor. For that, I got 9 options. 


So I guess my final result is  $$\frac{9}{4!}\Rightarrow 0.375$$
$37.5\%$ that none of the side neighbors knows the other side neighbor.
Is this correct? How would I calculate the count of all possible seating options where no side neighbor knows each other?
Thanks

A slight improvement on user8734617's method:
Exactly $\frac{1}{7}$ of the possible arrangements have a fixed pair $A_1,B_1$ seated together. To see this, given any arrangement with the pair seated together, fix the position of $A_1$ and rotate the other $7$ around the table, giving $6$ other arrangements with $A_1,B_1$ not seated together. This method produces all possible arrangements around the table, so each arrangement with $A_1,B_1$ seated together corresponds to $6$ other arrangements, meaning $A_1,B_1$ sit together with probability $\frac{1}{7}$.
By the same method, of those arrangements with a fixed pair $A_1,B_1$ seated together, $\frac{1}{5}$ have another fixed pair $A_2,B_2$ also seated together, so two fixed pairs sit together with probability $\frac{1}{7 \cdot 5}$; similarly, three fixed pairs sit together with probability $\frac{1}{7 \cdot 5 \cdot 3}$ and four with probability $\frac{1}{7\cdot5\cdot3\cdot1}$.
Then we apply inclusion-exclusion: our answer is $$\begin{align}
&1 - \binom{4}{1}\cdot\frac{1}{7} + \binom{4}{2}\cdot\frac{1}{7 \cdot 5} - \binom{4}{3}\cdot\frac{1}{7 \cdot 5 \cdot 3} + \binom{4}{4}\cdot\frac{1}{7\cdot5\cdot3\cdot1} \\
&= 1 - \frac{4}{7} + \frac{6}{7\cdot5} - \frac{4}{7 \cdot 5 \cdot 3} + \frac{1}{7 \cdot 5 \cdot 3} \\
&= \frac{3}{7} + \frac{6}{7\cdot5} - \frac{3}{7 \cdot 5 \cdot 3} \\
&= \frac{3}{7} + \frac{5}{7\cdot5} \\ &= \frac{4}{7}
\end{align}$$
