Estimate the number of ants in a colony A friend of mine gave me this weird problem I cannot solve. _To estimate the number of ants in a colony an entomologist draws 5500 ants randomly from the colony, labels them with a radioactive isotope and then put them back in. The day after he comes back to the colony and draws 70000 ants randomly. Given that 2 ants show the trace of the isotope, which is the estimated number of ants in the colony? (Assume no ant can die or be born)._ So there are 5500 special balls in an urn of $N$ balls. If I draw 70000 balls, the probability of finding $x$ special balls is (If I am not mistaken) $P(x)=\dfrac{\binom{5500}{x}\binom{N-5500}{70000-x}}{\binom{N}{70000}}$ should I find the max of this function (how?) and then use $x=2$ to find $N$? Or there are better ways?

I would run the simple proportion: $$ 2/70000=5500/N $$ The concentration of marked ants is about the same in big samples. So $N=5500*70000/2=192500000$