TITLE: Probability - Couples randomly sitting at a table, calculate the probability that they are together
QUESTION [6 upvotes]: I'm currently practicing for my first actuarial exam and came across this problem.  The posted solution doesn't make sense to me, and even if I'm right I don't know the correct way to do it.
The problem: 13 married couples are seated randomly at a round table.  Calculate E(X), where X is the number of husbands sitting next to their wives.
The given solution: Consider an individual couple.  The probability that that couple is seated together is $\frac 2 {25}$, so E(X) = $13(\frac 2 {25})$ = $\frac {26} {25}$
Me: What?  These aren't independent events!  I'm going to brute force a smaller version of this problem...
So I decided to tackle the problem for 2 couples instead of 13.  This gives us 24 permutations, 17 of which have both couples sitting together (X=2) and the rest of which have none (X=0).  Therefore E(X) = $\frac {34} {24}$
Using the solution from above, $2 (\frac 2 3) = \frac 4 3$.
To repeat my actual question: I'm pretty sure the given solution is wrong but I don't know what right is, so I'm looking for either an explanation for the flaw in my reasoning or the correct answer.
EDIT: OK, I rechecked my work and found my error.  There are actually 16 permutations making the answer for N=2 $\frac {32} {24} = \frac 4 3$.  I'll be off to bed now.

REPLY [5 votes]: The given solution is correct. Let us recall that:

The expectation of a sum of random variables is the sum of the expectations, whether these random variables are independent or not. 

In your case, $X$ is the sum of the $X_i$, where $X_i=1$ if husband and wife of couple $i$ are seated next to each other, and $X_i=0$ otherwise. True, the random variables $X_i$ are not independent, but the only important things are that, for every $i$, $\mathrm E(X_i)=\frac2{25}$, and that there are $13$ couples. Hence indeed $\mathrm E(X)=13\times\frac2{25}$.