So the probability of two poisson events taking place against time can be given as:
$$P = r_1 r_2 \int_0^\inf dt_1 e^{-r_1t_1} \int_{t_1 - T}^{t_1 + T} dt_2 e^{-r_2t_2}$$
Where $$ X = t_1 - t_2 \quad -T < X < T$$
Are limits set by the fact these two events need to be T seconds from each other. I want to extrapolate this to N independent events, where there's now a probability of no events taking place. I've started with the equation
$$P(k) = nCc \cdot (\Pi_{0}^k r_ie^{-r_it_i})(\Pi_k^N e^{-r_iT}) = ABC$$
Where A is the binomial coefficient, B is the distribution above made for the k events which have succeeded, and C is for the (N-k) that have not. If each event has the same rate, this simplifies a little:
$$P(k) = nCc \cdot (\Pi_{0}^k re^{-rt_i})e^{-r(N - k)T}$$
And this is where I'm at. I know it's possible to use a substitution of a lot of substitutions like the first expression to solve this, but I'm not sure how to do this when every event is within the same time window. Does anyone have any advice? I'm also happy to approximate a little as long as the answer would be close for high N