How should one combine the uncertainties from the ATLAS and CMS measurements? 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?
 A: Excellent question! I'm not the best person to give an answer, but here's what I think I know: all the detectors actually can run simultaneously. The LHC beams travel in a circle, so they are continually crossing, but normally the beams are so diffuse that they pass through each other with essentially no interactions. Around the detectors, though, there are focusing magnets which narrow the beams and increase the cross-sectional density so the number of collisions goes way up within the detector. Even so, the number of collisions that occur as the beams cross within a detector is not enough to make a significant difference in the beam luminosity from one detector to the next. For this reason, it seems doubtful that there will be correlations in the results from different detectors due to the effect of each detector on the beam. And in practice the ability to collect more data by running multiple experiments simultaneously is more important than eliminating any correlations that do exist.
It is possible that the results from different detectors would be correlated due to properties of the beam itself, but such correlations would be there whether the experiments run one at a time or not. 
As I understand it, these correlations make the process of combining the results from ATLAS and CMS rather nontrivial, along with differences between the design of the detectors. There is a working group at CERN that aims to produce a combined analysis of the ATLAS+CMS data, but their process takes several months. However, it is possible to produce a rough estimate. I imagine the procedure for doing so basically amounts to something like multiplying probabilities, but I'm not particularly familiar with it, so I will refer you to Philip Gibbs' posts at the viXra blog which are basically the canonical source for unofficial ATLAS+CMS combination plots.
A: Have a look at the recent discussions where the experiments are combined:
http://motls.blogspot.com/2012/02/higgs-signal-grew-from-38-to-43-sigma.html
and error estimates discussed :
http://blog.vixra.org/2012/02/10/some-technical-points-about-combining-sigmas/
Here are the recent individual ones:
http://www.science20.com/quantum_diaries_survivor/atlas_and_cms_publish_2011_higgs_results-86735
