Ten fish are caught in a lake, marked, and then returned to the lake. Two days later $20$ fish are again caught, two of which have been marked.
Find the probability of two of the $20$ fish being marked if the lake has $k$ fish (assuming the fish are caught at random).
we need to take $2$ fishes from marked $10$ fishes, which can be done in ${10 \choose 2}$ ways. Then rest $18$ fishes will be chosen from rest $k-10$ unmarked fishes. Therefore the probability is
$$\frac{{10 \choose 2}{k-10 \choose 18}}{k \choose 20}$$
But the problem asks for Which value of "$k$" the probability will be maximum.
I am totally stuck as this function becomes very lengthy to solve.
Help appreciated :)