There are $2m$ persons forming $m$ couples who live together at a given time. Suppose that at some later time, the probability of each person being alive is $p$, independently of other persons. At that later time, let $A$ be the number of persons that are alive and let $S$ be the number of couples in which both partners are alive. For any number of total surviving persons $a$, find $E[S∣A=a]$.
I don't have the answer to the problem. Here is my proposed solution, can anyone verify whether it is correct.
Let $$(S| A= a) = (S_1| A = a) + (S_2| A = a) + ... + (S_m | A = a)$$
Here $(S_i| A = a)$ is a indicator variable indicating whether both members in a couple are alive or not. So the probability of $p(S_i | A = a)$ is given by $$\frac{p(S_i \cap A = a)}{p(A = a)}$$ where the numerator is given by $p^2 {n - 2 \choose a-2}p^{a-2}(1-p)^{n-2-(a-2)}$ and denominator is given by ${n \choose a}p^a(1-p)^{n-a}$. Therefore $E[S_i | A = a]$ is now known to us, and by symmetry all the indicator variables will have same expectation. So we have the final answer as $m \cdot \frac{p(S_i \cap A = a)}{p(A = a)}$. If I am wrong, point out from the specific place where I went wrong.
Thanks!