Joint red and green Poisson count formula ($R\sim \mathrm{Poisson}(rm),G\sim \mathrm{Poisson}((1-r)m)$)

For a finite-intensity set of mean $m$, independent red marking with probability $r$ gives counts $R,G$ withDisjoint source sets give independent pairs of these counts. Refining overlapping test sets into disjoint cells then proves independence of the two whole Poisson point processes, not only their counts on one set.