Question.  Which of the above three examples will give better results if the goal is to obtain a numerical approximation of  [Graphics:Images/MonteCarloPiMod_gr_125.gif]


Consider the following ratios for examples 2, 3, 4.


The "estimate" for the accuracy of the estimated areas will be


Let's see how to use this formula for the results of examples 1, 2, and 3.



Hence the simulation values in example 4 should be a little closer to  [Graphics:../Images/MonteCarloPiMod_gr_132.gif]  than the simulation values in examples 2 and 3.  

Observation.  Notice that [Graphics:../Images/MonteCarloPiMod_gr_133.gif]  when  [Graphics:../Images/MonteCarloPiMod_gr_134.gif].  This is a consequence of  [Graphics:../Images/MonteCarloPiMod_gr_135.gif]  "filling up"  [Graphics:../Images/MonteCarloPiMod_gr_136.gif].    








































(c) John H. Mathews 2005