 Solution

With immediate release and ideal independent uniform sampling of fish, each catch is a Bernoulli trial with tagged-success probability $p=k/N$. Stopping at the $r$th success makes the count of untagged catches a negative binomial random variable. A total of $x$ failures requires $r-1$ successes among the first $x+r-1$ trials and a success on the last trial, so the probability is $\left(\genfrac{}{}{0ex}{}{x+r-1}{r-1}\right)p^r(1-p{)}^x$. Separate days use disjoint independent sequences of trials, hence their counts form an independent and identically distributed sample. Equal catch probabilities alone do not mathematically imply independence; this is the usual repeated independent sampling assumption in the stated model.