# Seating arrangements: Question about the book solution and summation indices

$$20$$ people are to be seated at seven tables, three of which have 4 seats and four of which have 2 seats. If the people are randomly seated, find the expected value of the number of married couples that are seated at the same table.

My question 1:

In the book solution (see screenshot below) I don't understand the last equality, specifically the indexes that go from $$i=1$$ to $$22$$ and $$19$$ respectively. It seems they should both go from $$i=1$$ to $$10$$.

My question 2:

Where am I going wrong in my approach?

Let $$X$$ be the number of married couples sitting at the same table. Let $$X_i = 1$$ if couple $$i$$ is sitting at the same table for $$i=1,...,10$$. Then

$$E[X] = E\left[\sum_{i=1}^{10} X_i \right] = \sum_{i=1}^{10} E\left[X_i \right] = 10 \cdot P(X_1 = 1)$$

where the last equality comes from LOE and symmetry. To find $$P(X_1 = 1)$$ I will condition on the event $$A =$$ the husband is at a table with $$4$$ seats and the event $$B =$$ the husband is at a table with $$2$$ seats.

$$P(X_1 = 1) = P(X_1 = 1 \mid A)P(A) + P(X_1 = 1 \mid B)P(B)$$

$$=\frac{3}{19}\frac{12}{20}+\frac{1}{19}\frac{8}{20} \approx .1157$$

and so $$E[X] \approx 1.157$$ which does not match the book solution of $$2.48$$. Where am I going off the rails?

Thanks for your help and patience.

Book solution • Problem is from Lord Sheldon Ross. I have seen a few typos but in almost every case when I think the book is wrong it's just me. – HJ_beginner Sep 23 '18 at 23:31
• Recently there's been quite a number of questions with heteronormative assumptions. I'm aware that books are full of these and we can't not use the books because of that, but I'd appreciate if people would at least comment on this offensive aspect when posting questions about such problems. A married couple does not consist of a wife and a husband. – joriki Sep 23 '18 at 23:35
• @joriki that's a good point... while I am an ignoramus when it comes to math I fortunately don't have a medieval view on love and marriage... though I suppose in a subtle way by posting these questions I inadvertently promote the idea that only a man and woman can marry... going forward I will do my best to stay mindful of this aspect – HJ_beginner Sep 23 '18 at 23:57
• In almost any other situation I'm pretty confident that I wouldn't make outdated assumptions about marriage. Like if I was addressing a group of people in real life or on another site like Reddit. As you point out in your meta post beginners should be given some benefit of the doubt... I know that as a newbie, I'd be paranoid that if I rephrased a word problem to eliminate offensive aspects I might accidentally alter the actual math question being asked. This is an unfortunate relic of the past... it's good that you're bringing this issue up to the greater community – HJ_beginner Sep 24 '18 at 0:50
• Good point about the rephrasing, as this is in fact a widespread problem on the site, rephrased and accidentally altered problems that require a lot of effort to clarify. – joriki Sep 24 '18 at 4:29

You're right on all counts. The limits on the sums are wrong, and if you correct them, the result in the book coincides with yours (except the correct result rounds to $$0.1158$$, not $$0.1157$$).