The sizes of animal populations are often estimated by using a capture-tag-recapture method. In this method $k$ animals are captured, tagged, and then released into the population. Some time later n animals are captured, and $Y$, the number of tagged animals among the $n$, is noted. The probabilities associated with $Y$ are a function of $N$, the number of animals in the population, so the observed value of $Y$ contains information on this unknown $N$. Suppose that $k=4$ animals are tagged and then released. A sample of $n=3$ animals is then selected at random from the same population. Find $P(Y=1)$ as a function of $N$. What value of $N$ will maximize $P(Y=1) ?$