I need to solve the following problem:
Given $n$ yes/no experiments, and a success probability $p$, at least how many successes $k$ can I expect (say, with a 'confidence' of $c=95\%$ or more)?
Another way to put it is: I want to be 95% sure to have at least $k$ successes. Everything is given, and I am interested in $k$.
As the Cumulative Distribution Function (CDF) for the Binomial Distribution is $$ F(k; n, p) = P(X \leq k) = I(1 - p; n - k, k + 1), $$ where $I(\cdot; \cdot, \cdot)$ denotes the regularized incomplete beta function, and I'm mostly interested in calculating the numerical outcome, the way I wanted to do it is to solve $$ P(X \geq k) = 1 - P(X \leq k - 1) = I(p; k, n - k + 1) = c $$ for $k$ using Maxima (although any other approach/tool would be good, too).
Doing some sanity checks with the CDF, though, I get unexpected results. For example, I would expect to get a probability value for
F(1; 100, .63) = beta_incomplete_regularized(.37, 99, 2)
However, Maxima outputs:
Any comments/hints to this output, or to the approach in general? What other tools would be simple to use here?