Negative Binomial Distribution
We use the Negative Binomial Distribution to answer the following question:
 Given a probability of success $p$ per trial, what is the probability that we got the $r$th success after $k$ trials, i.e. $P(X=k)$?

Support of $X={r,r+1,r+2,…}$

p.m.f. of $X$: $P(X=k)=(r−1k−1 )p_{r}(1−p)_{k−r}$ where

$k$ is the number of trials

$r$ is the number of successes

$p$ is the probability of success for 1 trial

$E(X)=pr $

$Var(X)=p_{2}r(1−p) $
Intuition
Consider the situation where you have $r$ = 3, $p=0.5$, and you want to find $k=7$.
Intuitively, this asks what are the odds that you get the 3rd success on the 7th try?
So for the first six tries, you have $(26 )$ options of successes, multiplied by $p_{2}$ odds of getting successes, and $(1−p)_{6−2}$ probability of failure, then a probability $p$ of getting the 7th try. Which yields the p.m.f. of the equation you see above.
Related
 A Negative Binomial Distribution is a sequence of Negative Binomial Distribution is a sequence of Geometric Distributions.