The generating function for the binomial distribution can be expressed as:
$((1-p)+p)^r=\sum\limits_{k\ge 0} \binom{r}{k}(1-p)^k p^{r-k}$
$=\sum\limits_{k\ge 0} \text{ probability of $k$ failures when stopping at $r$ trials}$
I was tinkering with the negative binomial distribution to understand the "negative" and obtained:
$((1-p^{-1})+p^{-1})^r=\sum\limits_{k\ge 0}\binom{-r}{k}(1-p^{-1})^k(p^{-1})^{-r-k}$
$=\sum\limits_{k\ge 0}\binom{-r}{k}(-1)^k(1-p)^kp^r=\sum\limits_{k\ge 0}\binom{r+k-1}{k}(1-p)^kp^r$
$=\sum\limits_{k\ge 0} \text{ probability of $k$ failures when stopping at $r$ successes}$
I'm curious if there is a non-algebraic (ideally combinatorial) motivation for the fact that mapping $p\mapsto p^{-1}$ maps the generating function for number of failures when stopping at $r$ trials to stopping at $r$ successes.