The Negative Binomial Distribution

The negative binomial models the number of trials  
\[n\]
  up to and including a given number of successes  
\[x\]
, where the probability of success is a fixed  
\[p\]
. The first  
\[x-1\]
  successes may be any of the first  
\[n-1\]
  trials, but the  
\[n\]
th trial must be the  
\[x\]
th success.
We may model the first  
\[x-1\]
  successes from  
\[n-1\]
  by the binomial  
\[B(n-1,p)\]
  distribution:
\[\begin{equation} \begin{aligned}P(X=x-1) &={}^{n-1}C_{x-1}p^{x-1}(1-p)^{(n-1)-(x-1)} \\ &={}^{n-1}C_{x-1}p^{x-1}(1-p)^{n-x}{}\end{aligned} \end{equation}\]

The  
\[n\]
th trial is the  
\[x\]
th success with probability  
\[p\]
, so the probability of the needing  
\[n\]
  trials to obtain  
\[x\]
  successes is  
\[{}^{n-1}C_{x-1}p^x(1-p)^{n-x}\]
.
It is important to realise that the variable being modelled here is  
\[n\]
.  
\[x\]
  is the fixed number of successes.
It is often convenient to reparametrize in terms of number of successes and failures, so let  
\[k\]
  be number of failures, then  
\[x+k=n\]
  and the expression becomes  
\[{}^{x+k-1}C_{x-1}p^x(1-p)^{k}\]
.