Congratulations on building a nice-looking GM tube cosmic ray counter array. The actual rate of cosmic rays is of the order of 1 count per minute per cm2 of active area, but it does depend on solid angle, so this number is only approximate. I will assume that each GM tube has a broadside active area about 10 cm2, so maybe 10 counts per minute is a good number.
On counting rate and accidental coincidences, it is important to record the singles counting rate for each of your tubes; S1, S2, and S3 counts per second.. If your coincidence gate width is τ seconds, then for example, your counter #1 is ON S1τ of the time, and you should have random councidences with tube #2 at a rate R12 = S1S2τ random coincidences per second. This random coincidence rate should be less than 1% of what you expect for the real coincidence rate, or about 0.1 counts per minute.
You should move the GM tubes away from your other instrumentation, to minimize noise pickup from transformers etc. Although your electromagnetic shielding looks good, it is not 100%. The GM tubes should also be shielded from one another. One reason is that discharges in one tube could affect another tube, but also some cosmic ray interactions in one tube could produce x-rays or gamma rays that can be detected by another. One example is Compton scattering, which produces a recoil gamma ray that could easily hit another tube. So putting a little x-ray shielding between the tubes; about an inch of aluminum, 1/2" copper, or 1/4" lead (but not iron), is a good idea. Another thing that is a lot more important for phototube scintillator systems than GM tubes is to alternate the directions the cables come out from each tube. You now have right, right, right. Using right, left, right is better.
I hope this helps.
Bob S
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