  A double quantum dot coupled to an optical cavity is a prototypical example
of a non-trivial open quantum system. Recent experimental and theoretical
studies show that this system is a candidate for single-photon detection in the
microwave domain. This motivates studies that go beyond just the average
current, and also take into account the full counting statistics of photon and
electron detections. With this in mind, here we provide a detailed analysis of
the waiting time statistics of this system within the quantum jump unravelling,
which allows us to extract analytical expressions for the success and failure
probabilities, as well as for the inter detection times. Furthermore, by
comparing single and multi-photon scenarios, we infer a hierarchy of occurrence
probabilities for the different events, highlighting the role of photon
interference events in the detection probabilities. Our results therefore
provide a direct illustration of how waiting time statistics can be used to
optimize a timely and relevant metrological task.
