Basically, I'd like to be able to answer questions in the form of "What is the number of trials needed to have at least $p_k$ probability of at least $k$ successes, given that on each trial the probability of success is $p_t$?".
Let $x$ be the number of trials, then the equation for the problem is: $$\sum_{i=k}^x \frac{p_t^k(1-p_t)^{x-k}x!}{i!(x-i)!}\geq p_k\tag{*}\label{*} $$ We can also determine lower bound for $x$:$$x>\frac{p_k}k{p_t}$$
After some googling, I discovered that this exact question has already been answered, but for small $k$'s normal approximation isn't applicable. I want to understand the problem deeper, so my questions are:
- There's no analytical solution for \eqref{*}, since gamma-function is a special function and can't be integrated. Is that correct?
- What's the best way to solve this numerically? May be there's another useful approximation or something more suitable than a simple root-finding algorithm.