I wonder a technique to extend the discrete probability space.
Here's an example from Concrete Mathematics EXERCISE 8.17:
Let $X_{n,p}$ and $Y_{n,p}$ have the binomial and negative binomial distributions, respectively, with parameters $(n,p)$. Prove that $\Pr(Y_{n,p}\le m) = \Pr(X_{m+n,p}\ge n)$.
The answer to the problem is also from Concrete Mathematics:
\begin{align} \Pr(Y_{n,p}\le m) &= \Pr(Y_{n,p}+n \le m+n) \\ &= \hbox{probability that we need $\le m+n$} \tag{1}\\ &= \hbox{probability that $m+n$ tosses yield $\ge n$ heads} \tag{2} \\ &= \Pr(X_{m+n,p}\ge n) \end{align}
Well, (1) and (2) are describing the same thing, but they're in different probability spaces, so we should extend these two probability spaces into a unique probability space, ensuring that the probability of each event doesn't change. How can we do it? I haven't a clear idea.
And the more general problem arises: How to extend a probability space? Is there any technique to do it, at least, treat part of problems?
Thanks for your help!