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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

1 Answer 1

up vote 4 down vote accepted

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

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