A nice example of how much effort MC simulations save is calculating the area of a shadow cast by a cylinder. If you orient it just right the answer is easy, bu

A nice example of how much effort MC simulations save is calculating the area of a shadow cast by a cylinder. If you orient it just right the answer is easy, but it's a properly taxing calculation to work out an analytical function by hand that works for any orientation. Seriously give it a try, it's much harder than it appears. You can make a spreadsheet in 3 minutes that numerically solves the problem with MC. Now imagine that the cylinder is partly transparent and you want to know how much light makes it through. A few minutes of adjustment and the spreadsheet will do it no problem. Now imagine there is another object embedded in the first with its own transparency. Again only a relatively small amount of time is needed to get the simulation working. The analytical solution is not reachable by mortals.