Probability of being closest to each other

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Suppose there are $N$ red dots and $M$ blue dots uniformly distributed in a square region with side $S$. Each red dot finds its closest blue dot and each blue dot finds its closest red dot. What is the probability that these closest peers will be mutual? For example, if $N_i = \mathrm{closest}(M_j)$ then the probability that $M_j = \mathrm{closest}(N_i)$?