First off, a naive theorist question - How are measurements divided between the different detectors at the LHC? I would imagine that for a short run time, say, the CMS detector is active and all the others are turned off (the beam just passes through) and the beam is directed so as to collide in the CMS detector. Then for another run period, ATLAS is activated and collisions happen only there, giving another set of data. This would mean that the measurements at ATLAS and CMS are completely independent. Is this correct?

Given that, maybe I missed it, but I never saw anyone quote a combined average and uncertainty for both the ATLAS and CMS excesses. If, let's say, there's a 1 in 300 chance that the CMS excess is a statistical fluke and 1 in 200 that the ATLAS excess is a statistical fluke, as someone with a very limited understanding of probability and statistics, I would just multiply the probabilities to see how likely it isn't a Higgs signal and looking at the small value, would be very excited. How does one go about rigorously calculating the combined likelihood that the excess is not a Higgs signal?