In Bernoulli trials with parameter $p$, let $N_n$ be the number of trials required to produce either $n$ successes or $n$ failures, whichever comes first. I would like to computer the probability distribution of $N_n$.
My solution is:
Suppose $S_n$ is the number of trials required to produce $n$ successes. Then $ P(S_n=k) = {{k-1}\choose{n-1}} (1-p)^{k-n}p^n.$
Suppose $F_n$ is the number of trials required to produce $n$ failures. Then $ P(F_n=k) = {{k-1}\choose{n-1}} p^{k-n}(1-p)^n.$
$N_n = \min\{S_n, F_n\}.$ But I doubt it is correct and even if it is correct, I am not sure how to proceed. Can I get some insights? Thanks!