How to compute 'pi' with a gun [Comment calculer 'pi' avec un fusil ]

How to compute 'pi' with a gun [Comment calculer 'pi' avec un fusil].




It suffices to use a square target with an inscribed circle inside. Then shoot randomly the target. In this picture there are NG=20692 shots inside the circle (Green points) and NR=5719 outside (Red points). The full square contains NG+NR=20692+5719=26411 points (Green and Red). Please note that two shots fired at the same point count as one and only one.

R being the circle radius, the areas of the circle and the square are respectively: 

                            2
                    C = pi*R
  
                             2
                    S = (2*R)

K being a certain constant, C and S can be approximated using the point numbers: 
                    
                    C ~ K*NG

                    
                    S ~ K*(NG+NR)
Hence: 
                                2
                     C      pi*R      pi
                    --- = -------- = ----
                     S          2     4
                           (2*R)
  
                            C          K*NG         20692
                    pi = 4*--- ~ 4*----------- = 4*------- = 3.13
                            S       K*(NG+NR)       26411

assuming a perfect random process...


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