Matching gender expectation 
If there are x men and y women and we pair them randomly (do not consider gender). What are the expected number of the pair man-man, man-woman, woman-woman respectively?

(Assume x,y large so that even or odd in total is neglectible, I just want an approximation)
Note: I am confusing. Why can we compute it just like the probability of drawing two balls out of the box with red balls and white balls inside? Many answers suggested that they are the same, but I think they are completely different, this one is like drawing two balls out of box repeatedly without putting back, and counting the numbers of each type.

$$(x+y)^2 = x^2 + 2 x y + y^2$$
Consequently, we expect $\frac{x^2}{(x+y)^2}$ male-male assignments, $\frac{2 x y}{(x+y)^2}$ male-female assignments, and $\frac{y^2}{(x+y)^2}$ female-female assignments.
