The main example is, you're considering leasing new equipment that might save you money. What's the risk that it will actually cost more, considering various ranges of potential numbers (and distributions)?
I think it's harder to apply to software since there are more unknowns (or the unknowns are fatter-tailed) but I still liked the book just for the philosophical framing at the beginning: you want to the measure things because they help you make decisions; you don't need perfect measurements since reducing the range of uncertainty is often enough to make the decision.
Talking purely in agile software development, there is an idea of "Flow Metrics" [1] to use for estimation, basically boiling it down to "How many stories can we finish in the next timeframe (Sprint or whatever), and what is the uncertainty associated with this number?" I really like the idea, but haven't been able yet to test it.
There may exist an analytical solution for this, but I wouldn't trust myself to derive it correctly. It would certainly be a huge mess.
If we add that the source is also a right cylinder instead of point source, and we want to add first order attenuation of emitted gammas by the source itself, the spreadsheet becomes only a bit more complex, but there will not be a pen and paper equation solution.
In this example every row of the spreadsheet would represent a hypothetical ray. One could randomly choose a location in the source, a random trajectory, and check if the photon intersects the detector. An alternative approach would be randomly choosing points in both target and detector, then doing additional math.
The results are recovered by making histograms and computing stats on the outputs of all the rows. You probably need a few thousand for most things at least. Remember roughly speaking 10k hits gets you ~1% statistics.
My understanding is that these sorts of questions come up in ML, and there are ways of dealing with it, but they can't converge nearly as fast as simple iterations like Newton's method. Even if I have to take a series approximation instead of a simple formula, I'll be able to use autodiff (or at worst, symbolic differentiation) to get quick and precise answers to these questions.
also in general bayesian statistics
random medium article: https://medium.com/pythoneers/monte-carlo-simulation-ideas-a...
On the other hand, if you find yourself running 1hour+ numpy simulations for days on end you might want to consider an analytical approach.