I know if I have the average $\lambda$ of the number of times an event occurs in a time interval and the number of occurred events $k$ then I can calculate the probability $p$ for a Poisson distribution.
To get the number of events that occurred for a certain probability $p$ while also having the average number of times an event occurs $\lambda$ since for this probability something is unusual and by knowing the number of events, I can observe and if the number of events occurred $k$ I am informed and can do something about it.
Now, the real question is: Is there is a way (an inverse equation, for example) that I can go for to calculate the number of events $k$ given $p$ and $\lambda$? Or this can only be done by using trial and error and by increasing the number of events $k$ until I reach the desired probability $p$? Thanks.