The purpose of the code was to make it clearer what is happening here. If someone is confused by the direct calculation, then they can also look at the simulati

The purpose of the code was to make it clearer what is happening here. If someone is confused by the direct calculation, then they can also look at the simulation and maybe things will become clearer. Also, you do need to have some actual insight into the problem to solve it directly with calculations, whereas you just have to model the situation to brute force a simulation and arrive at essentially the same answer (with a lot more compute used, but that's basically free now!).