#
# The program that approximates the probability of
# throwing two points into a 1x1 square, having distance
# less than r
#
two_pts:=proc(  r::numeric,
                n::integer
              )
   local x1, y1, x2, y2, i, count, dist;

   count:=0;

   for i to n do

      # simulation
      # the first point
      x1:=real_ran(0,1);
      y1:=real_ran(0,1);
      # the second point
      x2:=real_ran(0,1);
      y2:=real_ran(0,1);

      # testing and counting
      dist:=sqrt((x1-x2)^2+(y1-y2)^2);
      if dist < r then
         count:=count+1;
      fi;

   od;

   evalf( count/n );

end;
       






     #
     # function that output a random real number in [a,b]
     #
     real_ran:=proc(
                    a::numeric,
                    b::numeric
                   )
        local x;
        x:=rand();
        evalf( a+ (b-a)*x/999999999999 );

     end;