Say I have a detector that tells me if one or more events has occurred in a given time interval (and the length of this time interval is fixed, i.e., not decided by me). I am also able able to push a button that will cause an event to happen within that time interval with high (but unknown) probability if condition A is true. Condition A is either always true or never true. This detector also picks up events that occur naturally according to a Poisson process with an unknown and potential high lambda value.
If I take N pairs of measurements with this detector, one measurement pressing the button and one measurement without pressing it, and I get n1 and n2 detections respectively, how do I calculate the likelihood that condition A is true?
Seems like a typical problem, but in what I've be able to find so far, the detector is able to count events, not just say whether one or more happened.