Because my maths is far weaker than my coding skills, I would have chosen simulation to give me a rough figure, rather than no answer; so the OPs simulation fascinated me when compared to the mathematical answer.

Sometimes simulation is your only option, so this is a good skill to have in that case. This is meant to show both approaches.

Applied mathematics and computational science are both just simulation a lot of the time

Often analytic solutions aren’t required for sufficient insight into most problems, despite their parsimoniousness/prettiness

just wait til you find out how LLMs work