The event occurs on a given trial with the probability $p$. What is the number $k$ of independent trials needed for the event to occur no less than $N$ times with probability $q$ ?
What I've managed to express so far is probability $q$ through $p$ and $N$ (is this correct?):
$q = p^{k} + C^{k-1}_{k}p^{k-1}(1-p) + … + C^{N}_{k}p^{N}(1-p)^{k-N}$
But would like to know how to analytically express what I need (I can solve it with numeric methods, of course, but I want a non-iterative formula).
Best Answer
This is not a solution, per se, but rather a heuristic you can use to narrow your search for the smallest number of trials you need. I do not think that there is an analytic solution, but rather just a large scale approximation which is quite useful.
Your trials can be viewed as a random variables which take the value $1$ with probability $p$ and the value $0$ with probability $1-p$. These have expected value $p$ and variance $p(1-p)$. By the central limit theorem, the sum of $k$ independent trials (i.e., the number of successful trials you've had) will be approximately normally distributed, with mean $kp$ and variance $kp(1-p)$. We want the probability that we have at least $N$ successes to be $q$ or greater. We can phrase this in terms of the standard error function.
Let $\mathcal N(\mu,\sigma^2)$ denote a normally distributed random variable with mean $\mu$ and variance $\sigma^2$. We want
$$q=P(\mathcal N(kp,kp(1-p))>N)=P(\mathcal N(0,kp(1-p))>N-kp)=P\left(\mathcal N(0,kp(1-p))>\frac{N-kp}{\sqrt{kp(1-p)}}\right).$$
However, $P(\mathcal N(0,1)>x)=\frac{1}{2}-\frac{1}{2}\operatorname{erf}(x/\sqrt{2})$, so we need to solve
$$\operatorname{erf}^{-1}(1-2q)=\frac{N-kp}{\sqrt{2kp(1-p)}}.$$
The left hand side is a constant which can be computed easily using many different computer packages. This reduces the problem to something more computationally feasible than computing the sum in the problem for various values of $k$.