How to calculate the probability of at least k events each of probability m in a set of N independent trials. I have a probability of an event $=p$. For example ($p = 0.01$).  I have $N$ (say $1000$) independent trials.  I seek to calculate the probability of having exactly $k$ ($k= 1 \dots N$) events of in the set.
I have that the probability of having $0$ events is $(1-p)^N$, which is to say the probability of getting no event $(1-p)$ in each of the $N$ trials.
After that I am not sure where to go.  Simulations give very reasonable answers - i.e. a more or less Gaussian centered at 10 (1% of 1000 using the examples given).  But I am stumped for a closed form solution.

Probability of having exactly $k$ events (occurence probability $p$) out of $N$ is
$$ \binom{N}{k} p^{k} (1-p)^{N-k}$$
See Binomial Distribution (and that indeed can be approximated with the Gaussian).
