= Solution
Setting the mean equation's derivative to zero gives
$$
\boxed{\widehat N=\frac\alpha\delta=\frac5{0.2}=25\ \text{animals}}.
$$
When the actual current occupancy is $n=25$, the next admission has <exponential distribution> rate five and the next discharge has rate $n\delta=5$. The <memoryless property> makes these independent competing clocks conditional on the current occupancy. Therefore
$$
\boxed{P(\text{discharge before admission}\mid X=25)=\frac{25(0.2)}{5+25(0.2)}=\frac12}.
$$
The rate comparison uses the actual stated occupancy. Knowing only the mean occupancy would not justify replacing the random $X$ inside $X\delta/(\alpha+X\delta)$ by its mean.
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