Formula for the Negative binomial inverse cumulative function

For example, how many times ($N$) do I need to flip a coin ($p=0.5$) to have a $P=90\%$ probability of having observed $20$ heads. I empirically found that I need $N=20+28=48$. Is it correct?

Is there an explicit formula for the Negative binomial inverse cumulative function?

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If you have MATLAB, see mathworks.com/help/toolbox/stats/nbininv.html – Shai Covo Jan 26 '11 at 1:23
I use Maple (statevalf,idcdf,negativebinomial) but I would like to know if there is an explicit formula. – Jean-Pierre Jan 26 '11 at 1:47

We know that $F(k) = P(X \leq k) = 1-I_{p}(k+1,r)$ where $I_{p}(k+1,r)$ is the incomplete beta function. So $0.9 = 1-I_{0.5}(k+1,r)$. From this, I believe you have enough information to get $k$ and $r$ and hence $n$.

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Thank you for your answer. I know how to compute the incomplete beta from k and r but I can not see how to compute k and r from the value of the incomplete beta. what did I miss? – Jean-Pierre Jan 26 '11 at 1:41