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.