Given three types of balls, red, green and blue. The number of each type of balls is denoted as R,G, and B. The total number of balls N is even. We (uniformly) randomly group balls into pairs (so we have N/2 pairs). What is the expected number of red-green pairs?
I tried to run simulations on this, but ideally I want to find a closed form formula. Any ideas?
Line the pairs up and label them. Let $X_k$ be the indicator random variable that the $k$-th pair is red-green (that is, $X_k$ equals one if it is or zero if it is not). The expected value of this random variable is therefore the probability that the $k$-th pair is red-green. All such random variables have identical distribution (although not independent), therefore their expectations are equal.$$\forall k\in\{1..N/2\}~.~\mathsf E(X_k)=\mathsf E(X_1)$$
Let $X$ be the count for red-green pairs. This will equal $\sum_{k=1}^{N/2} X_k$ and expectation is linear, so the expectation for this count is:$$\begin{align}\mathsf E(X)&=\mathsf E(\sum_{k=1}^{N/2}X_k)\\[0.5ex]&\vdots\end{align}$$