Peter - 

I'll just throw out a few thoughts, in case one or two might help. 

Your best course of action often depends upon your goal.  Are you trying to explain a 'disruption,' or prove that something was done to cause it?  Are you talking about one kind of event which is repeated, or different kinds of events in a certain combination?  If you just want to know if a certain event, repeated, or set of circumstances, will likely cause a disruption, then you could look at the number of similar cases, and see how often they result in that problem, or you may need to run simulations, if possible.  (See "contingency tables" mentioned below.) 

But it sounds like you may need to reconsider the nature of random events.  Random events do not result in a uniform distribution. You should expect clusters.  Even if you estimate the probability of a certain combination or number of events to occur in a 12 hour period is low, that may not be useful information.  You would still expect that to happen at some time ... randomly.  But if an improbable set of circumstances in any 12 hours always (or often) results in a 'disruption,' and never (or seldom) do disruptions occur otherwise, then you might want to know this.  (What if part of these events occurred at the end of the last 12-hour period, and some at the beginning of the next?) 

A caution: If you design a hypothesis test to look into this, note that a lone p-value is nearly meaningless.  You need a type II error analysis, or something else, to investigate effect size. 

See the following:

Press release for the American Statistical Association:

http://www.amstat.org/newsroom/pressreleases/P-ValueStatement.pdf

 https://www.researchgate.net/publication/262971440_Practical_Interpretation_of_Hypothesis_Tests_-_letter_to_the_editor_-_TAS

You might look at the probability of this situation occurring and having this disruption, and the probability of not having this event, and not having this disruption.  You could set up a "contingency table."  I haven't looked at those in a long time, but you could look it up.  I think many years ago that I developed a power analysis for a Fisher 'Exact' Test, but I don't recall what I did with it.  And please remember the caution about the oft misused p-value.  But to look at the above probabilities, you might want to start with a contingency table.

This might be a case where graphics can help to visualize whatever it is you want to investigate. Graphics often help. 

Also, remember that these events may not be independent.  Even the same kind of event may be influenced by something that changes the probability of multiple occurrences.  

You mentioned the Poisson distribution, and your problem may relate more to reliability theory or quality control, so you may want to consider all of those topics.

What struck me when I first saw your question though, was to note (or remind you) that random does not mean uniform.  You should expect some  clustering for random events, and over enough time, you should expect to see the unusual.  It is likely that the unlikely will occur, on occasion.

Cheers - Jim

PS -  Ah.  I see you mention regression in your list of topics for this question.  That might be a good idea.  You could also look at the square root of the estimated variance of the prediction error for cases of interest.  It would be best to validate with independent data.  The problem I imagine, however, with whatever you do, might be too little data to have much 'confidence' in results.  -  As for SPSS, I have not used it in many years, and don't have any suggestions regarding that.  

Article Practical Interpretation of Hypothesis Tests - letter to the...

thank you for your response. i will review it. I think the biggest take away message is "random does not mean uniform." I need to digest this some more. thank you!

This is an interesting question. I must admit that I am not qualified to answer, but I would feel bad if I didn't commend the uncommon clarity with which you phrased the question. I'd love to hear how this works out.