# Probability - Couples randomly sitting at a table, calculate the probability that they are together

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.

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You might want to check that $17$ permutations in which both couples are sitting together. With $2$ couples, there is essentially one configuration (not counting rotations): husbands sitting across the table from their wives. – Dilip Sarwate Jan 23 '12 at 2:47
@DilipSarwate I'm starting to think I misinterpreted the question. I was interpreting it so that anyone could be sitting in any seat and I was expected to find how many husbands had their wives on their immediate left or right (a far harder problem!) – Chad Miller Jan 23 '12 at 3:13

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 $\mathrm E(X)=13\times\frac2{25}$.

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Just found my mistake; now I feel silly. Thanks for the help! – Chad Miller Jan 23 '12 at 3:53