= 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 distribution>[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 $\binom{x+r-1}{r-1}p^r(1-p)^x$. 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.
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