When it comes to probability theory I am an embarrasment. I want to try an answer and help you out if I could. But I really have no idea what I am doing. The only reason why I am responding is because my mathematical intutition tells me that what I wrote below is what you want. However, I really have absolutely no idea what identically independently distribution is, or any of that sort of stuff. So if I end up embarrasing myself then I promise never to post in the advanced probability section again .

I think this is referred to as thenegative binomial.

Among independent trials you want to compute the probability that after (where , is a fixed number) trails you have got exactly successes. Each success in this trial has probability . Therefore, you want to show .

In order for the success to happen at the trail it means there were success in the first trails. Ah, but that is the standard binomial distribution and so that gives us . But youalsowant to have a success on the trial, by independence, you need to multiply this expression by again. And this gives you exactly what you want to prove.