Suppose we sample 10 data points where 4 are deemed successes. The likelihood of this given the probability of success $p$ is $p^4(1-p)^6.$
Consider now the problem of continuously sampling until we reach 4 successes. We find that after 10 samples, the $10^{th}$ happened to be the $4^{th}$ success. What is the likelihood of this?
I realize that the second case is negative binomialy distributed i.e the probability of k successes where the $n^{th}$ trial is a success is given by ${{n-1}\choose{k-1}}p^k(1-p)^k$. Maybe I can somehow deduce the likelihood from this?