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]

Answer.

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

        [Graphics:../Images/MonteCarloPiMod_gr_126.gif]

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

        [Graphics:../Images/MonteCarloPiMod_gr_127.gif].

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

[Graphics:../Images/MonteCarloPiMod_gr_128.gif]



[Graphics:../Images/MonteCarloPiMod_gr_129.gif]
[Graphics:../Images/MonteCarloPiMod_gr_130.gif]
[Graphics:../Images/MonteCarloPiMod_gr_131.gif]

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