Before I pose my question I just want to note that I know that this question here been asked here before but the responses were only partially helpful.
A total of $2n$ people, consisting of $n$ married couples, are randomly seated (all possible orderings being equally likely) at a round table. Let $C_i$ denote the event that the members of couple $i$ are seated next to each other, $i = 1, ... , n$.
$(a)$ Find $P(C_i)$.
$(b)$ For $j \neq i$, find $P(C_j|C_i)$.
$(c)$ Approximate the probability, for $n$ large, that there are no married couples who are seated next to each other.
The answer for $(a)$ is $\frac{2}{2n-1}$ and for $(b)$ $\frac{2}{2n-2}$. Allegedly, the answer for $(c)$ is approximately $e^{-1}$ and one acquires that result via Poisson approximation but I am not convinced of this since we're not considering independent trials here. Admittedly, dependence appears to be quite weak but how do I know that it is negligible?
Edit: Perhaps my take on it makes no sense, but I need help solving (c) which is what I was trying to say.
