Overlap Probability of Empirical Distributions I have a load versus capacity problem, and I'm trying to determine the likelihood of failure. Here's a simple example of what I mean. 
I have load and capacity discrete pdfs that were determined through a series of computer calculations that analyzed different input values (Monte Carlo simulations to address uncertainties). The figures below are a simplified version of the output:


As you can see, there is some amount of overlap between the two. I am trying to determine the probability that the system will fail (i.e., the load with exceed the capacity).
Is this as simple as integrating the overlapping area of the empirical cdfs and multiplying, or am I looking for something more complicated, like a Bhattacharyya coefficient? 
 A: Unfortunately, I don't think you can really answer this question accurately without knowing a bit more about what drives load and capacity variability.
Consider the case of load and capacity in the electrical grid.  In the middle of the day in the summertime, as businesses have all their lights turned on and everybody is running their air conditioning, the load (the number of megawatts that people are demanding) soars.  On the other hand, at 3:00 a.m. on generally any day of the year, as almost everyone is asleep, load falls.  In spite of this fluctuating load problem, in developed countries at least, you almost never see a situation where load exceeds capacity (which would manifest as either a brownout or a rolling blackout).  Why is this?
It's because in the electrical grid, load and capacity are tightly correlated over time.  Engineers deliberately start up and shut down power plants, or subsections of power plants, in anticipation of time-varying changes in demand.
If your systems behaves in a similar way, it's not enough to simply calculate the overlap between the two overall accumulated static distributions.  Instead, you need to Monte Carlo both factors simultaneously, and look for individual instances (i.e., individual "rolls" of the Monte Carlo dice) where load exceeds capacity.  This is because, if the random factors in the Monte Carlo lead to correlated variability in both the load and capacity (or for that matter, anti-correlated variability as well) the probability of system failure will be different than just a simple pdf overlap calculation would suggest.
