I have an event $E$ that happens in some small timestep $\delta t$ with probability $\delta p(t')=\delta t\alpha(t')$. I'm trying to figure out the probability that the event happens at least once from $t'=0$ to $t'=T$.
I know that if $\alpha$ were time independent, then I would just look for the product of probabilities that $E$ does not happen at each timestep, i.e. \begin{equation} P_\text{at least once}(T)=1-\lim_{\delta t \to0}\left(1-\delta t\alpha\right)^{T/\delta t}=1-e^{-\alpha T} \end{equation} But when $\alpha$ has time dependence I don't have an exponent of $T/\delta t$ in my limit, I just have a product of $1-\delta p(t')$ at different timesteps, with different $\delta p(t')$s.
Am I missing some really basic way to do this? I have $\alpha(t')$ explicitly, but it's very ugly in the problem I'm studying, so I'm looking to see if there's a general solution for an arbitrary $\alpha(t')$.
Thanks in advance!