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Given k successes, I'm looking for the probability mass function for n the number of trials that occurred. Wikipedia and other sources give it as:

$ \Pr(X = n) =\binom{n-1}{k-1} p^k(1-p)^{n-k}$

I'm wondering why the coefficient is $\binom{n-1}{k-1}$ and not $\binom{n}{k}$. If we know there are k successes, why are we figuring out how to arrange k-1 of them?

If the answer is that we assume the last trial is a success, and that therefore there are only k-1 successes to arrange over the remaining n-1 trials, is there a different formula for the condition that the last trial isn't necessarily a success?

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The negative binomial does assume that the last observed value (when you stopped) is a success. This link might be helpful:https://stats.stackexchange.com/questions/45276/distribution-of-number-of-bernoulli-trials-to-a-given-number-of-successes

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