Solved – Relation between binomial and negative binomial

Based on binomial distribution, event ${X geq r}$ is the set of outcomes that satisfy "$n$ trials led to $r$ successes or more", which is equivalent to "$r$-th success happened at $n$-th trial or before", which is in turn equivalent to "$n$ trials or less were required to get $r$ successes", and that is it. $$begin{align*} P{X geq r} &= P{mbox{at least r successes in n trials}}\ &= P{mbox{r-th success in n-th trial or before}}\ &= P{mbox{n or fewer trials to get r successes}}\ &= P{Y leq n} end{align*}$$

The second relation is the complement of first relation that is: $$begin{align*} P{X geq r} &= P{Y leq n},\ 1 – P{X geq r} &= 1 – P{Y leq n},\ P{X < r} &= P{Y > n}\ end{align*}$$

The second relation means:

$$P{mbox{less than r successes in n trials}}= P{mbox{more than n trials to get r successes}}$$