Which one is the correct mean formula of negative binomial distribution? $\frac{r}{p}$ or $\frac{(1-p) r}{p}$?

Question

If we are looking to find the probability of observing the $6th$ head after $12$ independent flips and we let $X$ be the random variable for the number of flips of an unbiased coin

I found that there are 2 formulas about negative binomial distribtuion:

1. $\frac{(1-p) r}{p}$
2. $\frac{r}{p}$

So which one is the correct formula of mean of negative binomial distribution? The answer displays that I should use second one, why?

Answer 1

Let's consider $Y$ discrete random variable which gives amount of failures till first success in binomial experiment. Then $EY=\frac{q}{p}$

But if you take $Z$ discrete random variable which gives amount of failures till $r$ success, then you will have $EZ = \frac{rq}{p}$

Answer 2

It depends on how you are defining the distribution. If you use the convention that the distribution counts the number of failures before the $r$th success, in a sequence of i.i.d. trials with probability of success $p$. Here, the mean is $\frac{(1-p)r}{p}$.

Your #2 may be from an alternate definition of the distribution, where you count the number of trials (not failures) until the $r$th success, in which case the mean is $\frac{r}{p}$.