I'm having troubles in solving this probabilty problem. A group of $n$ players is given; they are divided into two teams with the following procedure. A number $X$ is chosen randomly from the set $\{1, 2, ..., n-1\}$ with all values equally likely. The remaining $n-X$ players form the blue team.
Among the players, I have to consider a particular player, player A. I need to find the probability that the team containing player A has size $k$ for $k = 1, 2, ... , n-1$.
This point is solved the following way: $$P(A\in R \cap R=k) + P(A\in B \cap B=k) = \frac{2k}{n(n-1)}$$.
I have problems with the following two points:
-Finding the mean of the size of the team containing player A
and
-After the two teams have been chosen, each team selects a captain, which is equally likely to be any member of that team. I need to find the conditional distribution of the size of the team containing player A, given that player A is the captain.
Can anybody show me how to tackle this problem? Thanks a lot!
(a) Let r.v. $Y$ be the size of the team with player $A$. You already found $P(y=k)$, so
$$E(Y) = \sum_{k=1}^{n-1}kP(Y=k) = \sum_{k=1}^{n-1} \dfrac{2k^2}{n(n-1)} = \dfrac{2n-1}{3} \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{using the identity $\sum_{j=1}^m j^2=\dfrac{m(m+1)(2m+1)}{6}$}.$$
(b) Define event $C = $ "Player $A$ is captain". Then, for $k=1,2,\ldots,n-1,$
\begin{align} P(Y=k\mid C) &= \dfrac{P(C\mid Y=k)P(Y=k)}{\sum\limits_{j=1}^{n-1} P(C\mid Y=j)P(Y=j)} \qquad\qquad\text{by Bayes Theorem} \\ & \\ &= \dfrac{\dfrac{1}{k}\dfrac{2k}{n(n-1)}}{\sum\limits_{j=1}^{n-1} \dfrac{1}{j}\dfrac{2j}{n(n-1)}} \\ & \\ &= \dfrac{1}{n-1}. \end{align}