An urn has m black balls. At each stage, a black ball is removed and a new ball that is black with probability $p$ and white with probability $1-p$ is put in its place. Find the expected number of stages needed until there are no more black balls in the urn.
The solution says that the number of stages is a negative binomial Random Variable with parameters $m$ and $1-p$ and so expectation is $\frac{m}{1-p}$ but I don’t understand how it is a negative bin. Random variable, can someone please explain that?
At the start there are $m$ black balls. Notice it states "at each stage, a black ball is removed" hinting at a white ball will never be removed.
So essentially we are seeing how long it takes to get $m$ successes, with a success being the white ball being put in the bag. (If a white ball is not selected, then we simply are replacing our black ball and we do not have a success.)
Perhaps it is clearer to see this way how we have a negative binomial distribution.