$20$ 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
Best Answer
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 $0.1157$).